New lower bounds on frequency estimation of a multitone random signal in noise

This paper deals with lower bounds on frequency estimation errors of a complex multitone random signal (i.e., a sum of complex exponentials with random amplitudes) in additive Gaussian noise from discrete time samples. We present simple closed-form expressions for lower bounds on the errors of an arbitrary estimator of the frequencies for any number of tones with unknown spectral levels. The bounds are expressed in terms of physical quantities from which some insight on the inherent limitations of the estimation problem is exploited. In particular, the contribution of the presence of other tones to the estimation error of the frequency of a certain tone is shown to be a function of the difference between the frequencies via the frequency response of the sampling window and its first derivative at these differences. For large amount of data (or large signal-to-noise ratio) the presented bound converges to the Cramer-Rao lower bound (CRLB) on the same estimation problem.