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

A multiplier (or mixer) cascaded with a low pass filter is the basic piece of any communication receiver. It is also used to register minute frequency fluctuations of an external oscillator (RF) under test versus the frequency of a local oscillator (LO). Such a scheme may also be viewed as the basic model of an electronic oscillator, with the amplifier noise at the RF input and the resonator signal at the LO input. We have investigated experimentally the whole spectrum of frequencies and amplitudes of beat signals and their frequency fluctuations at the IF output of the multiplier+filter. We have found evidence that all the dynamics follows arithmetical rules. As for the frequency of the beat signal it is defined from a diophantine approximation of the frequency ratio of input oscillators. As for the amplitude it is defined globally from the position of resolved fractions with respect to the uniform graduation. As for the frequency fluctuations a transition from white frequency noise to 1/f frequency noise is observed close to resonance. It is explained on the basis of number theory in relation to the Riemann problem concerning the distribution of prime numbers