THE ACHIEVABLE ACCURACY IN ESTIMATING THE INSTANTANEOUS PHASE AND FREQUENCY OF A CONSTANT AMPLITUDE SIGNAL

The paper explores the achievable accuracy in estimating the instantaneous phase and frequency of complex constant amplitude signals. It is based on modeling of 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, we derive the Cramer-Rao bound (CRB) of the instantaneous phase and frequency of constant amplitude polynomial-phase signals. Then we examine some properties of the CRB's and use these properties to estimate the order of magnitude of the bounds. Finally, we extend the analysis 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. Using the maximum likelihood estimates we 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.