Fast nearly ML estimation of the parameters of real or complex single tones or resolved multiple tones

This paper presents new computationally efficient algorithms for estimating the parameters (frequency, amplitude, and phase) of one or more real tones (sinusoids) or complex tones (cisoids) in noise from a block of N uniformly spaced samples. The first algorithm is an interpolator that uses the peak sample in the discrete fourier spectrum (DFS) of the data and its two neighbors. We derive Cramer-Rao bounds (CRB's) for such interpolators and show that they are very close to the CRB's for the maximum likelihood (ML) estimator. The new algorithm almost reaches these bounds. A second algorithm uses the five DFS samples centered on the peak to produce estimates even closer to hit. Enhancements are presented that maintain nearly ML performance for small values of N. For multiple complex tones with frequency separations of at least 4 pi/N rad/sample, unbiased estimates are obtained by incorporating the new single-tone estimators into an iterative "cyclic descent" algorithm, which is a computationally cheap nonlinear optimization. Single or multiple real tones are handled in the same way. The new algorithms are immune to nonzero mean signals and (provided N is large) remain near-optimal in colored and non-Gaussian noise.