$1/f$ 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.

Figure 1

Sample signal constructed from the sequence of nonoverlapping pulses (red curve): ${\tau }_q$ respective gap durations, ${\theta }_q$ respective pulse durations, $t_k$ respective pulse occurrence time, $a$ height of the rectangular pulses.

Figure 2

Power spectral density of the signal when pulse and gap durations are sampled from exponential distribution. Red curve corresponds to a numerical simulation conducted with $a=1$ and ${\theta }_c={\tau }_c=1$ (or, alternatively, ${\gamma }_{\theta }={\gamma }_{\tau }=1$). Black curve corresponds to Eq. (20).

Figure 3

Comparison of the power spectral densities in the $\alpha =1$ case. Red curve shows the case with the comparatively long exponential pulse durations (simulated with $a=1, {\theta }_c={10}^3$), while green curve shows the case with the comparatively short exponential pulse durations (simulated with $a={10}^3, {\theta }_c={10}^{-3}$). Black curves correspond to Eqs. (31) (solid) and (30) (dashed). Other simulation parameters: ${\tau }_{\text{min}}=1$ and ${\tau }_{\text{max}}={10}^4$.

Figure 4

Power spectral densities of a signal with rectangular pulses obtained sampling pulse durations from various distributions: exponential (red curve), degenerate (green), uniform (blue), and bounded Pareto (magenta) distributions. Gap durations were sampled from bounded Pareto distribution with $\alpha =1, {\tau }_{\text{min}}=1$ and ${\tau }_{\text{max}}={10}^4$. Dashed black lines have $1/f$ slope. Other simulation parameters: $a=1$ and ${\theta }_c={10}^3$ (with exponential distribution), $a={10}^{-1}$ and ${\theta }_c={10}^2$ (degenerate distribution), $a=3, {\theta }_{\text{min}}=0$ and ${\theta }_{\text{max}}={10}^3$ (uniform distribution), $a=3, {\alpha }_{\theta }=1, {\theta }_{\text{min}}=1$, and ${\theta }_{\text{max}}={10}^4$ (bounded Pareto distribution).

Figure 5

Comparison of the power spectral densities in the weakly nonergodic case. Different curves correspond to different observations times: $T={10}^4$ (red circles), ${10}^6$ (green triangles), ${10}^8$ (blue squares), and ${10}^{10}$ (magenta diamonds). Gap durations were sampled from the Pareto distribution (with $\alpha =1$ and ${\tau }_{\text{min}}=1$), unit size ($a=1$) pulse durations were sampled from the exponential distribution (with ${\theta }_c={10}^2$). Dashed black line has $1/f$ slope.