OFFPRINT Ramanujan sums analysis of long-period sequences and 1/f noise

Europhysics Letters (EPL) has a new online home at www.epljournal.org Take a look for the latest journal news and information on: • reading the latest articles, free! • receiving free e-mail alerts • submitting your work to EPL Abstract – 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, quasi-periodic 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 α spectra of the Fourier transform. Introduction. – Signal processing of complex time-varying series is becoming more and more fashionable in modern science and technology. Indices arising from the stock market, changing global climate, communication networks such as the Internet etc. are widely used tools for business managers or governmental representatives. There already exists a plethora of useful approaches for signal processing of complex data. The oldest and perhaps most widely used method is the Fourier analysis and its " fast " implementation: the fast Fourier transform (or FFT). Other complementary techniques such as wavelet transforms, fractal analysis and autoregressive moving average models (ARIMA) were developed with the aim of identifying useful patterns and statistics in otherwise seemingly random sequences [1]. Ramanujan sums are defined as power sums over primitive roots of unity. One can use an orthogonal property of these sums (closely related to the orthogonal property of trigonometric sums) to form convergent expansions of some arithmetical functions related to prime number theory [2,3]. Following the ideas of Gadiyar and Padma [4], the first author proposed to expand the domain of application of Ramanujan sum analysis from number theory to arbitrary real time series and introduced the concept