FURTHER SIMPLE APPROXIMATIONS TO THE CRAMER-RAO LOWER-BOUND ON FREQUENCY ESTIMATES FOR CLOSELY SPACED SINUSOIDS

It is demonstrated that the Cramer-Rao lower bound on frequency estima tes for a data record containing two closely-spaced cisoids in complex white Gaussian noise can be approximated by an extremely simple nonma trix expression. It extends earlier a work by explicitly retaining the difference in initial phases as a parameter of interest. The approxim ation to the bound is shown to have a root-mean-square error of about 10%, with occasional peak errors of about +/-25% over a wide range of data lengths and for frequency separations down to about one-tenth of the Rayleigh resolution limit. Further, it is demonstrated that the sa me basic form of the approximation handles the related cases of (a) fr equency estimation of a single real sinusoid in real noise and (b) fre quency estimation for a closely-spaced pair of real sinusoids in real noise.