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 estimates for a data record containing two closely-spaced cisoids in complex white Gaussian noise can be approximated by an extremely simple nonmatrix expression. It extends earlier work by explicitly retaining the difference in initial phases as a parameter of interest. The approximation 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 same basic form of the approximation handles the related cases of (a) frequency estimation of a single real sinusoid in real noise and (b) frequency estimation for a closely-spaced pair of real sinusoids in real noise