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

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.