Cramer-Rao Lower bounds for time delay and Doppler shift estimation

Several Cramer-Rao lower bounds for time delay and Doppler shift estimation have been obtained for different models by using various methods, but the researches are not systematic and extensive. In this paper, starting with a general model, Cramer-Rao lower bounds for time delay estimation, Doppler shift estimation, and joint time delay and Doppler shift estimation are derived by using Fisher information matrix. The present results are consistent with the previous ones; moreover, they are applicable to all signal-to-noise ratios. The bounds obtained under different conditions are compared and the relations between them are offered. Further explanations about the relations are also given from the point of view of information theory. Simulations are provided to support the theoretical results.