In applied mathematics, the regressive discrete Fourier series (RDFS) is a generalization of the discrete Fourier transform where the Fourier series coefficients are computed in a least squares sense and the period is arbitrary, i.e., not necessarily equal to the length of the data.
The one-dimensional RDFS proposed by Arruda (1992a) can be formulated in a very straightforward way.
Given a sampled data vector (signal)
The above equation can be written in matrix form as The least squares solution of the above linear system of equations can be written as: where
can be obtained from: The two-dimensional, or bidimensional RDFS proposed by Arruda (1992b) can also be formulated in a straightforward way.
Here the equally spaced data case will be treated for the sake of simplicity.
The general non-equally-spaced and arbitrary grid cases are given in the reference (Arruda, 1992b).
Given a sampled data matrix (bi dimensional signal)
one can write the algebraic expression: The above equation can be written in matrix form for a rectangular grid.
For the equally spaced sampling case :
we have: The least squares solution may be shown to be: and the smoothed bidimensional surface is given by: where
can be easily implemented analogously to the one-dimensional case (Arruda, 1992b).
Recently, a package that includes one and two-dimensional RDFS was developed in order to make easier its use in the free and open source software R: