An average Cramer-Rao bound for frequency offset estimation in frequency-selective fading channels

Several variations of Cramer-Rao bounds for carrier frequency offset (CFO) estimation in frequency-selective fading channels have been used to benchmark practical estimators´ performance or to design training signals for CFO estimation. Among them, the extended Miller-Chang bound (EMCB) provides a tighter bound than the CRB for locally unbiased estimators. However, there is no closed-form expression of the EMCB for the CFO estimation in frequency-selective fading channels with an arbitrary training signal. In this letter, we derive a closed-form exact average CRB (the EMCB) valid for any training signal and any signal structure for the CFO estimation over frequency-selective Rayleigh fading channels with uncorrelated or arbitrarily correlated taps. The accuracy and generality of the proposed average CRB expression, and its advantages over the existing expressions are corroborated by numerical and simulation results.