On the frequency and amplitude spectrum and the fluctuations at the output of a communication receiver

A mixer cascaded with a low-pass filter is the basic piece of any communication receiver. The frequency and amplitude fluctuations of beat signals recorded at the output of the mixer plus filter set-up are investigated experimentally and explained on the basis of number theory in relation to the Riemann problem concerning the distribution of prime numbers. The frequency of the beat signal is obtained from a diophantine approximation of the frequency ratio of the input oscillators. The amplitude is defined globally from the position of resolved fractions with respect to an equally spaced graduation. Time series analysis methods show that frequency fluctuations present a transition from white frequency noise to 1/f frequency noise close to resonance; the latter is compatible with an underlying fractal attractor. The same transition is observed in the case of time series computed from continued fraction expansions.