Designing second-order Costas loops and PLL´s to track Doppler shift-analysis and optimization

For coherent communications under Doppler shift conditions, the time it takes the receiver synchronization circuit to lose lock becomes a major concern. Design parameters optimization for both the Costas and PLL second-order carrier tracking loops under Doppler shift is presented for the case of a passive (imperfect integrator) loop filter. The optimization is made possible by using an extension of the Kramers (1940) expression for the mean time to lose lock (MTLL), which is an explicit function of both the Doppler shift and the phase jitter variance