  A heuristic procedure based on novel recursive formulation of sinusoid (RFS)
and on regression with predictive least-squares (LS) enables to decompose both
uniformly and nonuniformly sampled 1-d signals into a sparse set of sinusoids
(SSS). An optimal SSS is found by Levenberg-Marquardt (LM) optimization of RFS
parameters of near-optimal sinusoids combined with common criteria for the
estimation of the number of sinusoids embedded in noise. The procedure
estimates both the cardinality and the parameters of SSS. The proposed
algorithm enables to identify the RFS parameters of a sinusoid from a data
sequence containing only a fraction of its cycle. In extreme cases when the
frequency of a sinusoid approaches zero the algorithm is able to detect a
linear trend in data. Also, an irregular sampling pattern enables the algorithm
to correctly reconstruct the under-sampled sinusoid. Parsimonious nature of the
obtaining models opens the possibilities of using the proposed method in
machine learning and in expert and intelligent systems needing analysis and
simple representation of 1-d signals. The properties of the proposed algorithm
are evaluated on examples of irregularly sampled artificial signals in noise
and are compared with high accuracy frequency estimation algorithms based on
linear prediction (LP) approach, particularly with respect to Cramer-Rao Bound
(CRB).
