  To understand methodological features of the detrended fluctuation analysis
(DFA) using a higher-order polynomial fitting, we establish the direct
connection between DFA and Fourier analysis. Based on an exact calculation of
the single-frequency response of the DFA, the following facts are shown
analytically: (1) in the analysis of stochastic processes exhibiting a
power-law scaling of the power spectral density (PSD), $S(f) \sim f^{-\beta}$,
a higher-order detrending in the DFA has no adverse effect in the estimation of
the DFA scaling exponent $\alpha$, which satisfies the scaling relation $\alpha
= (\beta+1)/2$; (2) the upper limit of the scaling exponents detectable by the
DFA depends on the order of polynomial fit used in the DFA, and is bounded by
$m + 1$, where $m$ is the order of the polynomial fit; (3) the relation between
the time scale in the DFA and the corresponding frequency in the PSD are
distorted depending on both the order of the DFA and the frequency dependence
of the PSD. We can improve the scale distortion by introducing the corrected
time scale in the DFA corresponding to the inverse of the frequency scale in
the PSD. In addition, our analytical approach makes it possible to characterize
variants of the DFA using different types of detrending. As an application,
properties of the DFA using moving average filtering are discussed.
