  This paper presents a new method for modelling periodic signals having an
aperiodic trend, using the method of variable projection. It is a major
extension to the IEEE-standard 1057 by permitting the background to be time
varying; additionally, any number of harmonics of the periodic portion can be
modelled. This paper focuses on using B-Splines to implement a piecewise
polynomial model for the aperiodic portion of the signal. A thorough algebraic
derivation of the method is presented, as well as a comparison to using global
polynomial approximation. It is proven that B-Splines work better for modelling
a more complicated aperiodic portion when compared to higher order polynomials.
Furthermore, the piecewise polynomial model is capable of modelling the local
signal variations produced by the interaction of a control system with a
process in industrial applications. The method of variable projection reduces
the problem to a one-dimensional nonlinear optimization, combined with a linear
least-squares computation. An added benefit of using the method of variable
projection is the possibility to calculate the covariances of the linear
coefficients of the model, enabling the calculation of confidence and
prediction intervals. The method is tested on both real measurement data
acquired in industrial processes, as well as synthetic data. The method shows
promising results for the precise characterization of periodic signals embedded
in highly complex aperiodic backgrounds. Finally, snippets of the m-code are
provided, together with a toolbox for B-Splines, which permit the
implementation of the complete computation.
