We present and analyze the simple analytically solvable model of 1/f noise, which can be relevant for the understanding of the origin, main properties and parameter dependencies of the flicker noise. In the model, the currents or signals represented as sequences of the random pulses, which recurrence time intervals between transit times of pulses are uncorrelated with the shape of the pulse, are analyzed. It is shown that for the pulses of fixed area with random duration, distributed uniformly in a wide interval, 1/f behavior of the power spectrum of the signal or current in wide range of frequency may be obtained.
Deep Dive into Modelling of 1/f noise by sequences of stochastic pulses of different duration.
We present and analyze the simple analytically solvable model of 1/f noise, which can be relevant for the understanding of the origin, main properties and parameter dependencies of the flicker noise. In the model, the currents or signals represented as sequences of the random pulses, which recurrence time intervals between transit times of pulses are uncorrelated with the shape of the pulse, are analyzed. It is shown that for the pulses of fixed area with random duration, distributed uniformly in a wide interval, 1/f behavior of the power spectrum of the signal or current in wide range of frequency may be obtained.
The origin and omnipresence of 1/f noise is one of the oldest problems of the contemporary physics. Since the first observation of the flicker noise in the currents of electron tubes by Johnson [1], fluctuations of signals and physical variables exhibiting behavior characterized by a power spectral density diverging at low frequencies like 1/f have been observed in a wide variety of systems [2]. The widespread occurrence suggest that some underlying mechanism might exist. However, a fully satisfactory explanation has not yet been found and the general theory of 1/f noise is still an open question.
A simple procedures of integration or differentiation of the convenient (white noise, Brownian motion or so) fluctuating signals do not yield in the signal exhibiting 1/f noise. There are no simple linear, even stochastic, differential equations generating signals with 1/f noise. Therefore, 1/f noise is often modeled as the superposition of Lorentzian spectra with a wide range of relaxation times [3]. Summation or integration of the Lorentzians with the appropriate weights may yield 1/f noise [4].
In many cases the physical processes can be represented by a sequence of random pulses. Recently, considering signals and currents as consisting of pulses we have shown [5][6][7] that the intrinsic origin of 1/f noise may be a Brownian motion of the interevent time of the signal pulses, similar to the Brownian fluctuations of the signal amplitude, resulting in 1/f 2 noise.
The model, proposed in [5,6], can be extended taking into account finite duration of the pulse. The spectrum of the signal, consisting of the pulse sequences which belong to the class of Markov process, was investigated in [8,9].
In this article we present a different model of pulses. We consider a signal consisting of a sequence of uncorrelated pulses. The shape of the pulses is determined by only one random parameter -pulse duration. We will show that by suitably choosing of the distribution of the pulse duration the 1/f noise can be obtained.
We will investigate a signal consisting from a sequence of pulses. We assume that:
(i) the pulse sequences are stationary and ergodic;
(ii) interevent times and the shapes of different pulses are independent.
The general form of such signal can be written as
where functions A k (t) determine the shape of individual pulses and time moments t k determine when a pulse occurs. The power spectrum is given by the equation
where T = t f -t i . Substituting Eq. (1) into Eq. ( 2) we have
We assume that functions A k (u) decrease sufficiently fast when |u| → ∞. Since T → ∞, the bounds of the integration in Eq. ( 3) can be changed to ±∞ . We also assume that time moments t k are not correlated with the shape of the pulse A k . Then the power spectrum is
After introducing the functions
and
the spectrum can be written as
Equation ( 6) can be further simplified assuming that the process is stationary. In the stationary case all averages can depend only on k -k ′ . Then
and
Equation ( 6) then reads
Changing the variables into k ≡ k ′ and q ≡ k -k ′ and changing the order of summation we obtain
Introducing N = k max -k min we have
where ν = lim
is the mean number of pulses per unit time.
When the sum N q=1 q Re χ q (ω)Ψ q (ω) converges and T → ∞ then the second term in the sum (9) vanishes and the spectrum is
When the shape of the pulses is fixed (k-independent) then the function Ψ k,k ′ (ω) does not depend on k and k ′ and, therefore, Ψ k,k ′ (ω) = Ψ 0,0 (ω). Then equation ( 6) yields the power spectrum
This is the spectrum of one pulse multiplied by the spectrum of the sequence of δ-shaped pulses S δ (ω). It has been shown [5,6] that the spectrum of such a sequence can exhibit 1/f -like behaviour in a broad frequency range if the interevent times τ k = t k -t k-1 follow an autoregressive process.
When the pulses are uncorrelated and
is the Fourier transform of the pulse
¿From Eq. ( 11) we obtain the spectrum
When the interevent times τ k = t k -t k-1 are random and uncorrelated then
¿From Eq. ( 15) we obtain
Here
If the occurrence times of the pulses t k are distributed according to Poisson process then the interevent time probability distribution is Ψ(τ ) = 1 τ e -τ τ . The characteristic function obeys the equality Re χτ (ω)
1-χτ (ω) = 0 and the spectrum is
We will investigate this case more deeply.
Let the only random parameter of the pulse is the duration. We take the form of the pulse as
where T k is the characteristic duration of the pulse. The value β = 0 corresponds to fixed height pulses; β = -1 corresponds to constant area pulses. Differentiating the fixed area pulses we obtain β = -2. The Fourier transform of the pulse (20) is
¿From Eq. ( 19) the power spectrum is
Introducing the probability density P (T k ) of the pulses durations T k we can write
If P (T k ) is a power-law distribution, then the expressions for the spectrum are similar for all β.
For small frequencies we ex
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