Ramanujan sums for signal processing of low frequency noise

An aperiodic (low frequency) spectrum may originate from the error term in the mean value of an arithmetical function such as Mobius function or Mangoldt function, which are coding sequences for prime numbers. In the discrete Fourier transform (and the FFT) the analyzing wave is periodic and not well suited to represent the low frequency regime. In its place we introduce a new signal processing tool based on the Ramanujan SUMS c_q(n), well adapted to the analysis of arithmetical sequences with many resonances p/q. The sums are quasiperiodic versus the time n of the resonance and aperiodic versus the order q of the resonance. New results arise from the use of this Ramanujan-Fourier transform (RFT) in the context of arithmetical and experimental signals.