1fnoise from the sequence of nonoverlapping rectangular pulses Aleksejus Kononovicius Bronislovas Kaulakys Institute of Theoretical Physics and Astronomy Vilnius University

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1/f noise from the sequence of nonoverlapping rectangular pulses

Aleksejus Kononovicius∗

, Bronislovas Kaulakys

Institute of Theoretical Physics and Astronomy, Vilnius University

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.

1 Introduction

Flicker noise, also 1/f noise or pink noise, is a phenomenon well–known for almost a century since it was ﬁrst

observed by Johnson in a vacuum tube experiment [1, 2]. Since then power–law scaling in the power spectral

density of 1/fβform (with 0.5.β.1.5) has been reported in diﬀerent experiments and empirical data sets

across varied ﬁelds of research [3–7], and, especially, in solids [8–10]. One of the peculiarities of 1/f noise is

that it is observed for low frequencies and no cutoﬀ frequency has been observed in many cases, e.g., 300 years’

worth of weather data [11] or a three–week experiment with semiconductors [12], no cutoﬀ frequency has been

observed [13]. In other cases, the cutoﬀ frequency can be observed [14–16], but 1/f noise is still observed over

a broad range of frequencies.

Given observations in various research ﬁelds, one would expect that a general explanation of 1/f noise is

due. However, even after almost a century after discovery, there is no generally accepted model of 1/f noise.

There are numerous diﬀerent modeling approaches some of them based on actual physical mechanisms within

the systems in question, while some approaches aspire to provide a more general explanation. Mathematical

literature is rich in true long–range memory models, such as fractional Brownian motion [17], ARCH family

models [18], and ARFIMA models [19]. In physics literature one most commonly will see 1/f noise being

obtained by appropriately summing Lorentzian spectra as in the McWorther model [20, 21]. Self–organized

criticality framework was also put forward as a possible explanation [22], as well as the memoryless nonlinear

response [23]. Our group has built various nonlinear stochastic processes to model 1/f noise in a variety of

scenarios and diﬀerent modeling frameworks: autoregressive inter–event time point processes [21,24], stochastic

diﬀerential equations [25,26] and agent–based models [27]. For a detailed review of works by our group see [28].

Our group, as well as others, have observed that nonlinear transformations of Markovian stochastic processes

can lead to spurious long–range memory processes [29–32]. These are completely diﬀerent approaches as the

true long–range memory models rely on nonlocal operators, while the models exhibiting spurious long–range

memory rely on locally nonlinear potentials, which often result in nonergodic or nonstationary behavior.

Here we will consider a diﬀerent model, one which is not aﬀected by the nonlinear transformations of amplitude

and thus reproduces 1/f noise not due to ﬂuctuations in amplitude but due to temporal dynamics. The

approach we take here is most similar to renewal theory models [33], and random telegraph noise models, as we

∗email: aleksejus.kononovicius@tfai.vu.lt; website: http://kononovicius.lt

arXiv:2210.11792v5 [cond-mat.stat-mech] 14 Mar 2023

model a system which abruptly switches between two states (“on” and “oﬀ”). Thus the signal generated has the

characteristic look of a telegraph signal or pulse sequence [34]. In [35], Halford suggested that 1/f noise could

be modeled by a sequence of well–behaved perturbations with power–law distributed durations. Heiden [36]

considered a sequence of pulses, with the coupling between pulse amplitude, duration, and the gap duration, and

showed that for ﬁxed time integral pulses (of any arbitrary shape) 1/fβnoise will be obtained when the pulse

duration is power–law distributed. In [37] an opposite problem was solved: reconstruction of pulse duration

distribution given power spectral density and the characteristic pulse shape. Schick and Verveen have reported

a grain ﬂow experiment in which 1/f noise was observed with a low–frequency cutoﬀ [14]. A theoretical model

of triangular pulse sequences was also proposed to explain the experimental results. The power spectral density

of a signal with “on” and “oﬀ” states was examined in [38]. The autocorrelation function of a random telegraph

signal with power–law distributed “on” and “oﬀ” durations was obtained in [39]. Exploration of the nonergodic

case has led to further exploration of age–dependence of observed statistical properties [40] and a proposed

solution to the cutoﬀ paradox [13]. Theoretical and empirical analysis of 1/f noise in random telegraph–like

signals remains an active object of research (for more recent examples see [41–47]). In [48, 49] considered a

combination of the random telegraph–like dynamics turning the Poisson process on and oﬀ as an explanation

for 1/f noise in semiconductors. [50–53] have consider the random telegraph–like noise in the blinking quantum

dots experiments, in some cases leading to the prediction and experimental observation of the aging eﬀects in

the power spectral densities. Therefore, this is a third kind of approach to the modeling of the long–range

memory phenomenon, which is local in the event–time space, but is observed as nonlocal due to the observation

occurring in the real–time space [26].

In this paper, we consider a sequence of nonoverlapping rectangular pulses and show that 1/f noise can be

obtained when gap durations are short in comparison to the characteristic pulse duration and are power–law

distributed. In Section 2 we provide a generalized derivation of an expression for the power spectral density of

the signal constructed from the sequence of nonoverlapping pulses. In Section 3 we examine the case when the

pulse and gap durations are sampled from the exponential distribution. In Section 4 we examine a case when

gap durations are sampled from a power–law distribution (bounded Pareto distribution is used for analytical

derivations and numerical simulation), and examine the conditions when pure 1/f noise can be observed. We

ﬁnd that the range of frequencies over which 1/f noise is observed does nontrivially depend on the characteristic

duration of the pulses. In Section 5 we explore the implications of ﬁnite observation time on the reported results,

which yields a weakly nonergodic process exhibiting 1/f noise with low–frequency cutoﬀ observable only for the

extremely low frequencies. A summary of the obtained results is provided in Section 6.

2 Power spectral density of the sequence of nonoverlapping pulses

We investigate a stochastic process generating a sequence of nonoverlapping pulses with random durations θk.

The pulses are separated by gaps of random duration τk. In the general case this stochastic process generates

a signal which is given by a sum over all pulse proﬁles Ak(t)when the respective pulse occurs at time tk:

I(t) = X

Ak(t−tk).(1)

Note that, we assume that Ak(s)may have nonzero values only during the pulse. Before the pulse starts, s < 0,

and after the pulse ends, s>θk,Ak(s)is assumed to be zero. The truncation of pulse proﬁles and the gaps

between the pulses ensure that the pulses never overlap or touch. As the pulses are nonoverlapping, tkis given

by a sum of previous pulse and gap durations:

tk=

k−1

q=0

(θq+τq).(2)

Here, for notational simplicity, we have chosen that θ0= 0. When calculating the power spectral density of the

signal we ignore this artiﬁcially introduced “zeroth” pulse. In Fig. 1 we have plotted a sample signal constructed

from the sequence of nonoverlapping pulses and highlighted the aforementioned quantities. Note that if we

allow pulses to be almost instantaneous (if we take θq→0limit), then we obtain a point process case.

Figure 1: Sample signal constructed from the sequence of nonoverlapping pulses (red curve): τq– respective gap

durations, θq– respective pulse durations, tk– respective pulse occurrence time, a– height of the rectangular

pulses.

The power spectral density of the signal I(t)is given by

S(f) = lim

T→∞ *2

TZT

I(t)e−2πift dt

2+= lim

T→∞ *2

TX

e−2πiftkFk(f)

2+,(3)

where the averaging h. . .iis performed over distinct realizations of the process, Tis the duration of the signal,

and Fk(f)is the Fourier transform of the k-th pulse proﬁle. For rectangular pulses, the Fourier transform is

given by

Fk(f) = Zθk

Ak(u)e−2πifu du=aZθk

e−2πifdu=ia

2πf e−2πif θk−1,(4)

but at this point, let us keep our derivation general until the rectangular shape of the pulses is relevant. Let us

split the expression for the power spectral density into two terms:

S(f) = lim

T→∞ *2

e2πif(tk0−tk)Fk(f)F∗

k0(f)+= lim

T→∞ *2

k|Fk(f)|2++

+ lim

T→∞ *2

T X

k0>k

e2πif(tk0−tk)Fk(f)F∗

k0(f) + X

k0<k

e2πif(tk0−tk)Fk(f)F∗

k0(f)!+=

=S1(f) + S2(f),(5)

so we can deal with them separately. The ﬁrst term trivially simpliﬁes to

S1(f) = 2¯νD|Fk(f)|2E,(6)

where ¯νis the mean number of pulses per unit time. If the process is ergodic, and the observation time is long,

then the mean value of ¯νcan be trivially obtained from the mean values of the pulse and gap durations, i.e.,

¯ν=1

hθi+hτi. For nonergodic processes, or if the observation time is short, then ¯νneeds to be deﬁned as an

empirical mean number, i.e., ¯ν=K/T (here Kis the number of observed pulses). The two sums in the second

term diﬀer only in the sign of their imaginary parts, thus the second term can be rearranged by considering

only the real part:

S2(f) = 4 Re "lim

T→∞ *1

k0>k

e2πif(tk0−tk)Fk(f)F∗

k0(f)+#.(7)

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摘要：

1=fnoisefromthesequenceofnonoverlappingrectangularpulsesAleksejusKononovicius*,BronislovasKaulakysInstituteofTheoreticalPhysicsandAstronomy,VilniusUniversityAbstractWeanalyzethepowerspectraldensityofasignalcomposedofnonoverlappingrectangularpulses.First,wederiveageneralformulaforthepowerspectraldens...

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1fnoise from the sequence of nonoverlapping rectangular pulses Aleksejus Kononovicius Bronislovas Kaulakys Institute of Theoretical Physics and Astronomy Vilnius University

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