Noise, Oscillators and Algebraic Randomness pp 265-287 | Cite as

# 1/*f* Frequency Noise in a Communication Receiver and the Riemann Hypothesis

## Abstract

A mixer cascaded with a low pass filter is the central element 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 mixer + filter set-up. We have found evidence that all the dynamics follows arithmetical rules. The frequency of the beat signal is defined from a diophantine approximation of the frequency ratio of input oscillators; the amplitude is defined globally from the position of resolved fractions with respect to the equally spaced graduation; and for the frequency fluctuations, a transition from white frequency noise to 1/*f* frequency noise is observed close to resonance. This is explained on the basis of number theory in relation to the Riemann problem concerning the distribution of prime numbers. More precisely it is shown that diophantine signal processing is at work in the receiver and that this may be understood from the Littlewood and Franel-Landau formulation of the Riemann hypothesis.

## Keywords

Zeta Function Frequency Noise Local Oscillator Riemann Hypothesis Diophantine Approximation## Preview

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