10 research outputs found
Ramanujan sums analysis of long-period sequences and 1/f noise
Ramanujan sums are exponential sums with exponent defined over the
irreducible fractions. Until now, they have been used to provide converging
expansions to some arithmetical functions appearing in the context of number
theory. In this paper, we provide an application of Ramanujan sum expansions to
periodic, quasiperiodic and complex time series, as a vital alternative to the
Fourier transform. The Ramanujan-Fourier spectrum of the Dow Jones index over
13 years and of the coronal index of solar activity over 69 years are taken as
illustrative examples. Distinct long periods may be discriminated in place of
the 1/f^{\alpha} spectra of the Fourier transform.Comment: 10 page
Doppler Spectrum Estimation by Ramanujan Fourier Transforms
The Doppler spectrum estimation of a weather radar signal in a classic way
can be made by two methods, temporal one based in the autocorrelation of the
successful signals, whereas the other one uses the estimation of the power
spectral density PSD by using Fourier transforms. We introduces a new tool of
signal processing based on Ramanujan sums cq(n), adapted to the analysis of
arithmetical sequences with several resonances p/q. These sums are almost
periodic according to time n of resonances and aperiodic according to the order
q of resonances. New results will be supplied by the use of Ramanujan Fourier
Transform (RFT) for the estimation of the Doppler spectrum for the weather
radar signal