On least squares discrete Fourier analysis of unequally spaced data

Waldemar Popiński¹

¹ Space Research Centre Polish Academy of Sciences Bartycka 18a 00-716 Warszawa, Poland <a href="https://orcid.org/0000-0001-9585-1040">ORCID: 0000-0001-9585-1040</a>

Applicationes Mathematicae, Tome 47 (2020) no. 2, pp. 207-224

Résumé

The problem of discrete Fourier analysis of observations at non-equidistant times using the standard set of complex harmonics $\exp (i2\pi kt)$, $t\in \mathbb {R}$, $k=0,\pm 1,\pm 2,\ldots ,$ and the least squares method is studied. The observation model $y_j = f(t_j) + \eta _j$, $j=1,\ldots ,n$, is considered for $f\in L^2[0,1]$, where $t_j\in [(j-1)/n,j/n)$, and $\eta _j$ are correlated complex valued random variables with $E_\eta \eta _j=0$ and $ E_\eta |\eta _j|^2=\sigma _\eta ^2 \lt \infty $. Uniqueness and finite sample properties of the observed function Fourier coefficient estimators $\hat c_k$, $k=0,\pm 1,\ldots ,\pm m$, where $m \lt n/(8\pi )$, obtained by the least squares method, as well as of the corresponding orthogonal projection estimator $\hat f_N(t)=\sum _{k=-m}^m\hat c_k\exp (i2\pi kt)$, where $N=2m+1$, are examined and compared with those of the standard Discrete Fourier Transform.

Détail

DOI : 10.4064/am2399-6-2020

Keywords: problem discrete fourier analysis observations non equidistant times using standard set complex harmonics exp mathbb ldots least squares method studied observation model eta ldots considered where j eta correlated complex valued random variables eta eta eta eta sigma eta infty uniqueness finite sample properties observed function fourier coefficient estimators hat ldots where obtained least squares method corresponding orthogonal projection estimator hat sum m hat exp where examined compared those standard discrete fourier transform