Nonmatrix Cramer-Rao bound expressions for high-resolution frequency estimators

Analytical expressions are derived for the Cramer-Rao (CR) lower bound on the variance of frequency estimates for the two-signal time-series data models consisting of either one real sinusoid or two complex sinusoids in white Gaussian noise. The expressions give the bound in terms of the signal-to-noise ratio (SNR), the number N of data samples, and a function dependent on the frequency separation and the initial phase difference between the two signal components of each model. The bounds are examined as the phase difference is varied, and the largest and smallest bound expressions and the corresponding critical values of the phase difference are obtained. The exact expressions are analyzed for the case of small frequency separations /spl delta//spl omega/. It is found that the largest bound is proportional to (N/spl middot//spl delta//spl omega/)/sup -4//N/sup 3//spl middot/SNR and that the smallest bound is proportional to (N/spl middot//spl delta//spl omega/)/sup -2/N/sup 3//spl middot/SNR for small /spl delta//spl omega/. Examples indicate that the small /spl delta//spl omega/ results closely approximate the exact ones whenever the frequency separation is smaller than the Fourier resolution limit. Based on the asymptotic results, it is found that the threshold SNR at which an unbiased estimator can resolve the two signal frequencies is at least proportional to (N/spl middot//spl delta//spl omega/)/sup -6//N for the worst phase difference case and to (N/spl middot//spl delta//spl omega/)/sup -4//N for the best phase difference case for small /spl delta//spl omega/. The results are applicable to the general case of sampling where the samples are taken at arbitrary instants.