A short survey on nonlinear models of the classic Costas loop: Rigorous derivation and limitations of the classic analysis

Abstract

Abstract-Rigorous nonlinear analysis of the physical model of Costas loop -a classic phase-locked loop (PLL) based circuit for carrier recovery, is a challenging task. Thus for its analysis, simplified mathematical models and numerical simulation are widely used. In this work a short survey on nonlinear models of the BPSK Costas loop, used for pre-design and post-design analysis, is presented. Their rigorous derivation and limitations of classic analysis are discussed. It is shown that the use of simplified mathematical models, and the application of non rigorous methods of analysis (e.g., simulation and linearization) may lead to wrong conclusions concerning the performance of the Costas loop physical model.

Tutorial on dynamic analysis of the Costas loop

“…Also the errors caused by numerical integration (e.g. in MATLAB and SPICE) can lead to unreliable results [28,29,50,51].…”

“…The transfer functions for the loop filter and for the VCO have been given in Eqs. (26) and (29), respectively.…”

“…Based on the non linear model shown in Figure 11 and in Eqs. (26), (29), and (88) we can create a differential equation for the instantaneous difference frequency ∆ω as a function of time. For this type of Costas loop the differential equation has the…”

Lock-in range of classical PLL with impulse signals and proportionally-integrating filter

“…By contrast, the equations of model in the signal's phase space are autonomous (Gardner, 1966;Shakhgil'dyan and Lyakhovkin, 1966;Viterbi, 1966), what simplifies the study of PLL-based circuits. The application of averaging methods (Mitropolsky and Bogolubov, 1961;Samoilenko and Petryshyn, 2004) allows one to reduce the model of PLL-based circuits in the signals space to the model in the signal's phase space (see, e.g., (Leonov et al, 2012;Leonov and Kuznetsov, 2014;Kuznetsov et al, 2015b,a;Best et al, 2015).…”

Lock-in range of PLL-based circuits with proportionally-integrating filter and sinusoidal phase detector characteristic

“…The models of classical Costas loop and two-phase Costas loop in the signal's phase space can be described by relations similar to (5) (PD characteristic of the circuits usually is a π-periodic function, and the approaches presented in this paper can be applied to these circuits as well) (see, e.g., (Best et al, 2014;Best et al, 2015)).…”