Carleman framework filtering of nonlinear noisy phase-locked loop system
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Prashant G. Medewar
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Shambhu N. Sharma
Abstract
In control and communications, the phase-locked loop (PLL) is regarded as the demodulator. Under the presence of small noise, the PLL system fails to accomplish the locking condition resulting in the stochastic phase difference. As a result of this, the PLL becomes a nontrivial nonlinear stochastic system. To circumvent the curse of dimensionality and nonlinearity, we exploit the method of linearization in the Carleman framework in combination with the finite closure for the stochastic system considered here. We show that the Carleman linearization has proven useful to preserve the nonlinearity via bilinearization. The Carleman setup of the nonlinear stochastic differential system has the Markov property and the terms are manageable. Then, we filter the states of the PLL using the filtering theory of the homogeneous Markov process. Finally, the numerical simulations reveal the superiority of the proposed filtering in Carleman setting in contrasts with the celebrated extended Kalman filtering framework.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: None declared.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] K. Nishiguchi and Y. Uchida, “Transient analysis of the second-order phase-locked loop in the presence of noise,” IEEE Trans. Inf. Theor., vol. 26, pp. 482–486, 1980. https://doi.org/10.1109/tit.1980.1056216.Suche in Google Scholar
[2] A. Blanchard, Phase-Locked Loops, in Application to Coherent Receiver Design, New York, Wiley, 1976.Suche in Google Scholar
[3] H. Hu and L. K. Oxenløwe, “Chip-based optical frequency combs for high-capacity optical communications,” Nanophotonics vol. 10, pp. 1367–1385, 2021. https://doi.org/10.1515/nanoph-2020-0561.Suche in Google Scholar
[4] W. C. Lindsey, Synchronization Systems in Communications and Control, Upper Saddle River, Prentice-Hall, 1972.Suche in Google Scholar
[5] Z. Cheng and G. Liu, Communication Electronic Circuits, Information and Computer Engineering, Berlin, Boston, De Gruyter, 2020.10.1515/9783110593822Suche in Google Scholar
[6] J. W. M. Bergmans, “Effect of loop delay on stability of discrete-time PLL,” IEEE Trans. Circ. Syst. Fund. Theor. Appl., vol. 42, no. 4, pp. 229–231, 1995. https://doi.org/10.1109/81.382480.Suche in Google Scholar
[7] P. Dupuis and H. J. Kushner, “Stochastic systems with small noise, analysis and simulation: a phase-locked loop example,” SIAM J. Appl. Math., vol. 47, pp. 643–661, 1987. https://doi.org/10.1137/0147043.Suche in Google Scholar
[8] A. J. Viterbi, “Phase-locked loop dynamics in the presence of noise by Fokker–Planck techniques,” Proc. IEEE, vol. 51, pp. 1737–1753, 1963. https://doi.org/10.1109/proc.1963.2686.Suche in Google Scholar
[9] B. G. Gawalwad and S. N. Sharma, “Coloured noise analysis of a phase-locked loop system: beyond Itô and Stratonovich stochastic calculi,” Differ. Eqn. Dyn. Syst., vol. 24, no. 2, pp. 231–245, 2016. https://doi.org/10.1007/s12591-014-0212-z.Suche in Google Scholar
[10] J. I. Statman and W. J. Hurd, “An estimator-predictor approach to PLL loop filter design,” IEEE Trans. Commun., vol. 38, no. 10, pp. 1667–1669, 1990. https://doi.org/10.1109/26.61435.Suche in Google Scholar
[11] T. Carleman, “Application de la theorie des equations integrals lineaires aux systemes d equations differentialles non Lineaires (in French),” Acta Math., vol. 59, pp. 63–87, 1932. https://doi.org/10.1007/bf02546499.Suche in Google Scholar
[12] J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Trans. Circ. Syst., vol. 25, no. 9, pp. 772–781, 1978. https://doi.org/10.1109/tcs.1978.1084534.Suche in Google Scholar
[13] W. L. Rugh, “Nonlinear system theory,” in The Volterra/Wiener Approach, Baltimore, Johns Hopkins University Press, 1981.Suche in Google Scholar
[14] R. Bellman and J. M. Richardson, “On some questions arising in the approximate solution of nonlinear differential equations,” Q. Appl. Math., vol. 20, no. 4, pp. 333–339, 1963. https://doi.org/10.1090/qam/144472.Suche in Google Scholar
[15] H. Kunita, “Itô’s stochastic calculus: its surprising power for applications,” Stoch. Process. their Appl., vol. 120, no. 5, pp. 622–652, 2010. https://doi.org/10.1016/j.spa.2010.01.013.Suche in Google Scholar
[16] K. Kawalski and W. H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization, Singapore, World Scientific, 1991.10.1142/1347Suche in Google Scholar
[17] B. B. Purkayastha and K. K. Sarma, A Digital Phase Locked Loop Based Signal and Symbol Recovery System for Wireless Channel, India, Springer, 2015.10.1007/978-81-322-2041-1Suche in Google Scholar
[18] B. K. Øksendal, Stochastic Differential Equations, in An Introduction with Applications, Berlin, Germany, Springer-Verlag, 1998.10.1007/978-3-662-03620-4_1Suche in Google Scholar
[19] A. H. Jazwinski, Stochastic Processes and Filtering Theory, New York, and London, Academic Press, 1970.Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Testing of logarithmic-law for the slip with friction boundary condition
- A new clique polynomial approach for fractional partial differential equations
- The modified Rusanov scheme for solving the phonon-Bose model
- Delta-shock for a class of strictly hyperbolic systems of conservation laws
- The Cădariu–Radu method for existence, uniqueness and Gauss Hypergeometric stability of a class of Ξ-Hilfer fractional differential equations
- Novel periodic and optical soliton solutions for Davey–Stewartson system by generalized Jacobi elliptic expansion method
- The simulation of two-dimensional plane problems using ordinary state-based peridynamics
- Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format
- Simulation of the crystallization processes by population balance model using a linear separation method
- PS and GW optimization of variable sliding gains mode control to stabilize a wind energy conversion system under the real wind in Adrar, Algeria
- Characteristics of internal flow of nozzle integrated with aircraft under transonic flow
- Magnetogasdynamic shock wave propagation using the method of group invariance in rotating medium with the flux of monochromatic radiation and azimuthal magnetic field
- The influence pulse-like near-field earthquakes on repairability index of reversible in mid-and short-rise buildings
- Intelligent controller for maximum power extraction of wind generation systems using ANN
- A new self-adaptive inertial CQ-algorithm for solving convex feasibility and monotone inclusion problems
- Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions
- A study on solvability of the fourth-order nonlinear boundary value problems
- Adaptive control for position and force tracking of uncertain teleoperation with actuators saturation and asymmetric varying time delays
- Framing the hydrothermal significance of water-based hybrid nanofluid flow over a revolving disk
- Catalytic surface reaction on a vertical wavy surface placed in a non-Darcy porous medium
- Carleman framework filtering of nonlinear noisy phase-locked loop system
- Corrigendum
- Corrigendum to: numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Testing of logarithmic-law for the slip with friction boundary condition
- A new clique polynomial approach for fractional partial differential equations
- The modified Rusanov scheme for solving the phonon-Bose model
- Delta-shock for a class of strictly hyperbolic systems of conservation laws
- The Cădariu–Radu method for existence, uniqueness and Gauss Hypergeometric stability of a class of Ξ-Hilfer fractional differential equations
- Novel periodic and optical soliton solutions for Davey–Stewartson system by generalized Jacobi elliptic expansion method
- The simulation of two-dimensional plane problems using ordinary state-based peridynamics
- Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format
- Simulation of the crystallization processes by population balance model using a linear separation method
- PS and GW optimization of variable sliding gains mode control to stabilize a wind energy conversion system under the real wind in Adrar, Algeria
- Characteristics of internal flow of nozzle integrated with aircraft under transonic flow
- Magnetogasdynamic shock wave propagation using the method of group invariance in rotating medium with the flux of monochromatic radiation and azimuthal magnetic field
- The influence pulse-like near-field earthquakes on repairability index of reversible in mid-and short-rise buildings
- Intelligent controller for maximum power extraction of wind generation systems using ANN
- A new self-adaptive inertial CQ-algorithm for solving convex feasibility and monotone inclusion problems
- Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions
- A study on solvability of the fourth-order nonlinear boundary value problems
- Adaptive control for position and force tracking of uncertain teleoperation with actuators saturation and asymmetric varying time delays
- Framing the hydrothermal significance of water-based hybrid nanofluid flow over a revolving disk
- Catalytic surface reaction on a vertical wavy surface placed in a non-Darcy porous medium
- Carleman framework filtering of nonlinear noisy phase-locked loop system
- Corrigendum
- Corrigendum to: numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
