Asymptotic and modified Cramer-Rao bounds for frequency estimation in parallel fading channels

Modified and asymptotic Cramer-Rao bounds (MCRB and ACRB, respectively) for frequency estimation in parallel fading channels are presented. The bounds are derived for arbitrary correlation of fading processes in parallel channels and covariance of additive Gaussian noise. The results are specified for additive white noise. Numerical comparison with the true Cramer-Rao bound (CRB) for multipath Rayleigh fading channels shows that the ACRB is close to the true CRB, while the MCRB is not. In a multiple-input single-output (MISO) Rician fading channel, the ACRB presents a tight lower bound for the maximum-likelihood (ML) frequency estimation error. In general, the results obtained can be applied to scenarios with a signal of interest represented as an expansion in a basis of linear independent functions with random expansion coefficients. Specifically, they are applicable to communications in multipath, multiantenna, and fast fading channels, as well as any combination of these channel types.