Lower bounds for estimation of frequency and phase of Doppler signals

In phase Doppler anemometry (PDA) the phase between two sinusoidal signals should be estimated. In this paper we calculate theoretical lower bounds for how accurately this can be done (Cramer-Rao lower bounds). We prove that the maximum likelihood estimator for phase is obtained by a modified cross spectrum calculation, and show that this reaches the Cramer-Rao bound. We investigate specifically the effect of quantization of the Doppler signals, both theoretically and through computer simulations. We show that 1 bit quantization gives very serious problems around certain frequencies, whereas 4 bits are adequate for frequency estimation and more than 4 bits are needed for phase estimation. The results in this paper are also applicable in other fields in which estimation of the frequency and phase (difference) of sinusoidal signals is used.