TY - JOUR

T1 - Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large

AU - Leonov, G. A.

AU - Kuznetsov, N. V.

AU - Yuldashev, M. V.

AU - Yuldashev, R. V.

N1 - Funding Information: This work was supported by Saint Petersburg State University, Russian Foundation of Basic Research (Russia) , and the Academy of Finland . The authors would like to thank R.E. Best (Best Engineering company, Oberwil, Switzerland) for valuable discussions on practical implementations of Costas loops.

PY - 2015/3

Y1 - 2015/3

N2 - The analysis of the stability and numerical simulation of Costas loop circuits for high-frequency signals is a challenging task. The problem lies in the fact that it is necessary to simultaneously observe very fast time scale of the input signals and slow time scale of phase difference between the input signals. To overcome this difficult situation it is possible, following the approach presented in the classical works of Gardner and Viterbi, to construct a mathematical model of Costas loop, in which only slow time change of signal's phases and frequencies is considered. Such a construction, in turn, requires the computation of phase detector characteristic, depending on the waveforms of the considered signals. While for the stability analysis of the loop near the locked state (local stability) it is usually sufficient to consider the linear approximation of phase detector characteristic near zero phase error, the global analysis (stability in the large) cannot be accomplished using simple linear models. The present paper is devoted to the rigorous construction of nonlinear dynamical model of classical Costas loop, which allows one to apply numerical simulation and analytical methods (various modifications of absolute stability criteria for systems with cylindrical phase space) for the effective analysis of stability in the large. Here a general approach to the analytical computation of phase detector characteristic of classical Costas loop for periodic non-sinusoidal signal waveforms is suggested. The classical ideas of the loop analysis in the signal's phase space are developed and rigorously justified. Effective analytical and numerical approaches for the nonlinear analysis of the mathematical model of classical Costas loop in the signal's phase space are discussed.

AB - The analysis of the stability and numerical simulation of Costas loop circuits for high-frequency signals is a challenging task. The problem lies in the fact that it is necessary to simultaneously observe very fast time scale of the input signals and slow time scale of phase difference between the input signals. To overcome this difficult situation it is possible, following the approach presented in the classical works of Gardner and Viterbi, to construct a mathematical model of Costas loop, in which only slow time change of signal's phases and frequencies is considered. Such a construction, in turn, requires the computation of phase detector characteristic, depending on the waveforms of the considered signals. While for the stability analysis of the loop near the locked state (local stability) it is usually sufficient to consider the linear approximation of phase detector characteristic near zero phase error, the global analysis (stability in the large) cannot be accomplished using simple linear models. The present paper is devoted to the rigorous construction of nonlinear dynamical model of classical Costas loop, which allows one to apply numerical simulation and analytical methods (various modifications of absolute stability criteria for systems with cylindrical phase space) for the effective analysis of stability in the large. Here a general approach to the analytical computation of phase detector characteristic of classical Costas loop for periodic non-sinusoidal signal waveforms is suggested. The classical ideas of the loop analysis in the signal's phase space are developed and rigorously justified. Effective analytical and numerical approaches for the nonlinear analysis of the mathematical model of classical Costas loop in the signal's phase space are discussed.

KW - Costas loop

KW - BPSK

KW - Phase-locked loop (PLL)

KW - Nonlinear analysis

KW - Phase detector characteristic

KW - Phase comparator

KW - Simulation

KW - Stability in the large

UR - http://www.scopus.com/inward/record.url?scp=84907720346&partnerID=8YFLogxK

U2 - 10.1016/j.sigpro.2014.08.033

DO - 10.1016/j.sigpro.2014.08.033

M3 - Article

VL - 108

SP - 124

EP - 135

JO - Signal Processing

JF - Signal Processing

SN - 0165-1684

ER -