$1/f$ noise from the sequence of nonoverlapping rectangular pulses

TL;DR

This paper derives a general formula for the power spectral density of signals made of nonoverlapping rectangular pulses and shows that pure 1/f noise can occur under specific conditions involving power-law distributed durations.

Contribution

It provides a new analytical framework for understanding 1/f noise in signals composed of nonoverlapping pulses with power-law distributed durations.

Findings

Pure 1/f noise observed at low frequencies with long characteristic pulse durations.

Results applicable to both ergodic and weakly nonergodic processes.

Analytical derivation of spectral density for nonoverlapping pulse sequences.

Abstract

We analyze the power spectral density of a signal composed of nonoverlapping rectangular pulses. First, we derive a general formula for the power spectral density of a signal constructed from the sequence of nonoverlapping pulses. Then we perform a detailed analysis of the rectangular pulse case. We show that pure $1/f$ noise can be observed until extremely low frequencies when the characteristic pulse (or gap) duration is long in comparison to the characteristic gap (or pulse) duration, and gap (or pulse) durations are power-law distributed. The obtained results hold for the ergodic and weakly nonergodic processes.