Steady state distribution of a hyperbolic digital tanlock loop with extended pull-in range for frequency synchronization in High Doppler environment

A hyperbolic arctan based digital tanlock loop (D-TLL) operating with complex signals at base-band or intermediate frequencies in high Doppler environments is treated here. The arctan based loop, known as the tanlock loop (TLL), is used in software defined radio architectures for frequency acquisition and tracking. The hyperbolic nonlinearity intentionally introduced within the phase detector extends the pull-in range of the frequency for a given loop, compared to the normal D-TLL, allowing a wider frequency acquisition range which is suitable for high Doppler communications environment. In this paper we study the steady state phase noise performances of such a feedback loop for additive Gaussian noise using stochastic analysis. The stochastic model of a first-order hyperbolic loop and the theoretical analysis for the corresponding statistical distribution of the closed loop steady state phase noise are presented. The theoretical results are also verified by simulations.