  In many processes, the variations in underlying characteristics can be
approximated by noisy multi-periodic patterns. If large-scale patterns are
superimposed by a noise with long-range correlations, the detection of
multi-periodic patterns becomes especially challenging. To solve this problem,
we developed a discrete double Fourier transform (DDFT). DDFT is based on the
equidistance property of harmonics generated by multi-periodic patterns in the
discrete Fourier transform (DFT) spectra. As the large-scale patterns generate
long enough equidistant series, they can be detected by the iteration of the
primary DFT. DDFT is defined as Fourier transform of intensity spectral
harmonics or of their functions. It comprises widely used cepstrum transform as
a particular case. We present also the relevant analytical criteria for the
assessment of statistical significance of peak harmonics in DDFT spectra in the
presence of noise. DDFT technique was tested by extensive numerical
simulations. The practical applications of DDFT technique are illustrated by
the analysis of variations in solar wind speed related to solar rotation and by
the study of large-scale multi-periodic patterns in DNA sequences. The latter
application can be considered as generic example for the general spectral
analysis of symbolic sequences. The results are compared with those obtained by
the cepstrum transform. The mutual combination of DFT and DDFT provides an
efficient technique to search for noisy large-scale multi-periodic patterns.
