Regressive discrete Fourier series

The one-dimensional RDFS proposed by Arruda (1992a) can be formulated in a very straightforward way. Given a sampled data vector (signal) ${\displaystyle x_{n}=x(t_{n})}$, one can write the algebraic expression:

${\displaystyle x_{n}=\sum _{k=-q}^{q}X_{k}e^{\frac {-i2\pi kt_{n}}{T}}+\varepsilon _{n},t_{n}{\text{ arbitrary }},\quad n=1,\dots ,N.\,}$

Typically ${\displaystyle t_{n}=n\,\Delta t}$, but this is not necessary.

The above equation can be written in matrix form as

${\displaystyle WX=x+\varepsilon .\,}$

The least squares solution of the above linear system of equations can be written as:

${\displaystyle {\hat {X}}=(W^{H}W)^{-1}W^{H}x\,}$

where ${\displaystyle X^{H}}$ is the conjugate transpose of ${\displaystyle X}$, and the smoothed signal is obtained from:

${\displaystyle {\hat {x}}=W{\hat {X}}\,}$

The first derivative of the smoothed signal ${\displaystyle {\hat {x}}}$ can be obtained from:

${\displaystyle {\frac {dx}{dt}}(t_{n})=\sum _{k=-q}^{q}{\frac {-i2\pi k}{T}}X_{k}e^{\frac {-i2\pi kt_{n}}{T}},\quad n=1,\dots ,N.\,}$

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) ${\displaystyle x_{mn}=x(\xi _{m},\nu _{n}),m=1,\dots ,M;\ n=1,\dots ,N;}$ one can write the algebraic expression:

${\displaystyle x_{mn}=\sum _{k=-p}^{p}\sum _{l=-q}^{q}X_{kl}e^{\frac {-i2\pi k\xi _{m}}{L_{\xi }}}e^{\frac {-i2\pi l\nu _{n}}{L_{\nu }}}+\varepsilon _{mn},\quad m=1,\dots ,M;\ n=1,\dots ,N.\,}$

The above equation can be written in matrix form for a rectangular grid. For the equally spaced sampling case :${\displaystyle \xi _{m}=m\Delta \xi ,\nu _{n}=n\Delta \nu \,}$ we have:

${\displaystyle x_{mn}=\sum _{k=-p}^{p}\sum _{l=-q}^{q}X_{kl}e^{\frac {-i2\pi km\Delta \xi }{L_{\xi }}}e^{\frac {-i2\pi ln\Delta \nu }{L_{\nu }}}+\epsilon _{mn},\quad m=1,\dots ,M;\ n=1,\dots ,N.\,}$

The least squares solution may be shown to be:

${\displaystyle {\hat {X}}=(W_{L_{\xi }}^{H}W_{L_{\xi }})^{-1}W_{L_{\xi }}^{H}xW_{L_{\nu }}^{*}(W_{L_{\nu }}W_{L_{\nu }}^{H})^{-1}\,}$

and the smoothed bidimensional surface is given by:

${\displaystyle {\hat {x}}=W_{L_{\xi }}{\hat {X}}W_{L_{\nu }}^{t}\,}$

where ${\displaystyle X^{H}}$ is the conjugate, and ${\displaystyle X^{t}}$ is the transpose of ${\displaystyle X}$.

Differentiation with respect to ${\displaystyle \xi {\text{ and }}\nu }$ can be easily implemented analogously to the one-dimensional case (Arruda, 1992b).