Advertisement

Differential Equations and Dynamical Systems

, Volume 24, Issue 2, pp 231–245 | Cite as

Coloured Noise Analysis of a Phase-Locked Loop System: Beyond Itô and Stratonovich Stochastic Calculi

  • Balaji G. GawalwadEmail author
  • Shambhu N. Sharma
Original Research
  • 182 Downloads

Abstract

The phase-locked loop (PLL) system is an appealing electronic circuit that synchronizes oscillator output signal with the reference signal. In theoretical studies, it is revealed that the PLL system fails to achieve the locking condition in the presence of small noise influences. For this reason, it is reasonable to accomplish noise analysis of the PLL system. In this paper, we accomplish the PLL noise analysis in coloured noise framework that is more general as well as confirms the real noise statistics in contrast to the white noise. The extended phase space and stochastic differential rules for the random state vector satisfying vector stochastic differential equation are exploited to accomplish the PLL coloured noise analysis. It is worthwhile to mention that the three concepts, non-linearity, stochasticity, multi-dimensionality, received attentions in applied mathematics literature. However, applications of these three mathematical concepts to engineering problems are relatively very scarce. More precisely, this paper accounts for non-linearity, non-Markovian stochasticity as well as multi-dimensionality in the noise analysis of the PLL dynamics. Since the non-linearity, multi-dimensionality are ubiquitous in real physical systems as well as stochasticity will set future research directions in applied mathematics, systems and control, this paper will have lasting influence on future stochastic problems.

Keywords

Extended state space Non-linearity Non-Markovian stochasticity Ornstein–Uhlenbeck (OU) process Second-order PLL systems Stochastic differential equations (SDEs) Stochastic differential rules 

Notes

Acknowledgments

The Authors express gratefulness to anonymous Reviewers for their finer comments on the manuscript. That helped improve the content of the present version of the paper.

References

  1. 1.
    Best, R.E.: Phase-Locked Loops. McGraw-Hill, New York (1984)Google Scholar
  2. 2.
    Blanchard, A.: Phase-Locked Loops: Application to Coherent Receiver Design. Wiley, New York (1976)Google Scholar
  3. 3.
    Blanchard, A.: Interferences in phase-locked loops. IEEE Trans. Aerosp. Electron. Syst. 10, 686–697 (1974)CrossRefGoogle Scholar
  4. 4.
    Brancik, L., Kolarova, E.: Analysis of higher-order electrical circuits with stochastic parameters via SDEs. Adv. Electr. Comput. Eng. 13(1), 17–22 (2013)CrossRefGoogle Scholar
  5. 5.
    Diaconis, P.: Application of the method of moments in probability and statistics, In: Landau, H. J. (ed.) Moments in Mathematics. Proceedings of Symposia in Applied Mathematics, American Mathematical Society 37, pp. 125–142 (1987)Google Scholar
  6. 6.
    Dupuis, P., Kushner, H.J.: Stochastic systems with small noise, analysis and simulation: a phase-locked loop example. SIAM J. Appl. Math. 47, 643–661 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gardner, F.M.: Phaselock Techniques. Wiley, New York (1979)Google Scholar
  8. 8.
    Gawalwad, B. G., Sharma, S. N.: On a perturbed phase-locked loop system: a simple physical model. 2013 Multi-Conference on Systems and Control, Hyderabad, 28–30 Aug (2013)Google Scholar
  9. 9.
    Kawanishi, S., Takara, H., Saruwatari, M., Kioth, N.: Ultra high-speed phase locked-loop-type clock recovery circuit using a traveling-wave laser diode amplifier as a 50 GHz phase detector. Electr. Lett. 29, 1714–1716 (1993)CrossRefGoogle Scholar
  10. 10.
    Kolarova, E.: An application of stochastic integral equations to electrical networks. Acta Electrotech. Inf. 8(3), 14–17 (2008)Google Scholar
  11. 11.
    Kudrewisz, J., Wasowicz, S.: Equations of Phase-Locked Loops. Dynamics on Circles. Tours and Cylinder, World Scientific, Upper Saddle River (2007)Google Scholar
  12. 12.
    Kushner, H.J.: Approximations to optimal non-linear filters. IEEE Trans. Autom. Control 12, 546–556 (1967)CrossRefGoogle Scholar
  13. 13.
    Lindsey, W.C.: Synchronization Systems in Communication and Control. Prentice-Hall, Upper Saddle River (1972)Google Scholar
  14. 14.
    Lindsey, W.C., Chie, C.M.: A survey of digital phase-locked loops. Proc. IEEE 69, 410–431 (1981)CrossRefGoogle Scholar
  15. 15.
    Mao, X.: Stochastic Differential Equations and Applications. Horwood Publishing, Chichester (1998)Google Scholar
  16. 16.
    Mehrotra, A.: Noise analysis of phase-locked loops. IEEE Trans. Circuits Syst. I 49, 1309–1316 (2002)CrossRefGoogle Scholar
  17. 17.
    Mumford, D.: The dawning of the age of stochasticity. In: Arnold, V.I., Atiyah, M., Lax, P., Mazur, B. (eds.) Mathematics: Frontiers and Perspectives, pp. 197–218. American Mathematical Society, Washington, DC (2000)Google Scholar
  18. 18.
    Murray, R.M., Astrom, K.J., Boyd, S.P., Brockett, R.W., Stein, G.: Future directions in control in an information-rich world. IEEE Control Syst. Mag. 23, 20–33 (2003)CrossRefGoogle Scholar
  19. 19.
    Nishiguchi, K., Uchida, Y.: Transient analysis of the second-order phase-locked loop in the presence of noise. IEEE Trans. Inf. Theory 26, 482–486 (1980)CrossRefzbMATHGoogle Scholar
  20. 20.
    Pederson, D.O., Mayaram, K.: Analog Integrated Circuits for Communication: Principles Simulation and Design. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  21. 21.
    Pugachev, V.S., Sinitsyn, I.N.: Stochastic Differential Systems (Analysis and Filtering). Wiley, New York (1987)zbMATHGoogle Scholar
  22. 22.
    Razavi, B.: Monolithic Phase-Locked Loops and Clock Recovery Circuits. Wiley-IEEE Press, New York (1992)Google Scholar
  23. 23.
    Risken, H.: The Fokker–Planck Equation: Methods of Solution and Application. Springer, Berlin (1984)zbMATHGoogle Scholar
  24. 24.
    Terdik, G.Y.: Stationary solutions for bilinear systems with constant coefficients. In: Cinlar, E., Chug, K.L., Getoor, R.K. (eds.) Seminar on Stochastic Processes, pp. 196–206. Birkhäuser, Boston (1989)Google Scholar
  25. 25.
    Viterbi, A.J.: Phase-locked loop dynamics in the presence of noise by Fokker–Planck techniques. Proc. IEEE 51, 1737–1753 (1963)CrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2014

Authors and Affiliations

  1. 1.Department of Electrical EngineeringNational Institute of TechnologySuratIndia

Personalised recommendations