The achievable accuracy in estimating the instantaneous phase and frequency of a constant amplitude signal

The approach is based on modeling the signal phase by a polynomial function of time on a finite interval. The phase polynomial is expressed as a linear combination of the Legendre basis polynomials. First, the Cramer-Rao bound (CRB) of the instantaneous phase and frequency of constant-amplitude polynomial-phase signals is derived. Then some properties of the CRBs are used to estimate the order of magnitude of the bounds. The analysis is extended to signals whose phase and frequency are continuous but not polynomial. The CRB can be achieved asymptotically if the estimation of the phase coefficients is done by maximum likelihood. The maximum-likelihood estimates are used to show that the achievable accuracy in phase and frequency estimation is determined by the CRB of the polynomial coefficients and the deviation of true phase and frequency from the polynomial approximations