A simple model for pink noise from amplitude modulations

We propose a simple model for the origin of pink noise (or 1/f fluctuation) based on the waves with accumulating frequencies. These waves arise spontaneously in a system with synchronization, resonance, and infrared divergence. Many waves with accumulating frequencies can produce signals of arbitrary small frequencies from a system of small size. This beat mechanism can be understood as amplitude modulation. The pink noise can appear after the demodulation process, which produces a variety of pink noise in many fields. The pink noise thus formed from the beat has nothing to do with dissipation or long-time memory. We also suggest new ways of looking at pink noise in earthquakes, solar flares, and stellar activities.


Introduction
Pink noise is ubiquitous.This noise is characterized by the power-law behavior in the very low-frequency region of the power spectrum density (PSD) with power −α, (0.5 < α < 1.5).This noise is also known as 1/f fluctuation or flicker noise.
Since the first discovery of pink noise in a vacuum tube current 1 , the same noise has been observed in many systems: semiconductors, thin metals, biomembranes, crystal oscillators, very long-term temperature variations, the loudness of orchestral music, fluctuations in the Earth's rotation speed, fluctuations in the intensity of cosmic rays, heartbeats, postural control, magnetoencephalography and electroencephalography in the brain, etc. 2,3 .
There have been many discussions about the origin of pink noise [2][3][4] , but there seems to be no clear conclusion.Many models have been proposed that give rise to pink noise, but no universal mechanisms have been discovered.
Since pink noise is ubiquitous, the mechanism should be simple enough.However, all the applications of the basic concepts and techniques of the standard statistical mechanics seem to have encountered conflicts and disputes.Then people have tended to consider more fundamental concepts that can rewrite the theory of standard statistical mechanics.
A typical mechanism for producing arbitrary low-frequency fluctuations would be the wave beat, or amplitude modulation, of the primary high-frequency fluctuations.This amplitude modulation would be successful for pink noise if the frequencies were more concentrated in a small range.Then the secondary beat wave can have lower frequencies.One of the authors has already proposed this mechanism for the pink noise of sounds and music 5 .
Furthermore, this concentration should be cooperative and systematic to form the power-law PSD.We propose at least three types of cooperative systems that can produce pink noise.They are a) synchronization (section three), b) resonance (section four), and c) the infrared (IR) divergence (section five).
If the pink noise were an amplitude modulation, the demodulation mechanism should also exist.This is because the entire modulated data has only high-frequency information, while the data after demodulation can explicitly show the low-frequency information, including the pink noise.The demodulation mechanism can be intrinsic to the system or it can be prepared in the measurement procedure.Many demodulation mechanisms make the pink noise phenomena diverse: taking the square of the original signal, rectification, thresholding, etc.For example, when the electrical current or voltage exceeds the threshold in the biological body, ignition occurs and produces spikes in the nerve cells.Thus the possible pink noise in the electric current is transferred to the nerve signal.
We begin our discussion in section two, listing crucial clues to the origin of pink noise, all of which point to the possibility that pink noise is amplitude modulation.We then propose three mechanisms that lead to the modulation.In section three, we discuss the most typical mechanism synchronization.We show that a) exponential synchronization yields a power index of −1, and power-law synchronization yields a power index slightly different from −1.In section four, b) resonance also yields pink noise since the concentration of the excited eigenmodes around the fiducial frequency is systematically approximated by the exponential function in the relevant domain.In section five, c) infrared divergence in the bremsstrahlung can give pink noise.In section six, we discuss the robustness of pink noise and several demodulation mechanisms that yield a variety of pink noise.In the final conclusion section seven, we summarize our proposal and possible verifications based on the points presented in section two.We also summarize our future prospects of amplitude modulation on a variety of systems.

Some crucial clues for pink noise
We will now list some crucial clues to the origin of the pink noise.This process is quite important, because it can clarify which principles of statistical mechanics are useful and which are not useful to describe the pink noise.

Wave
Systems that exhibit pink noise are often waves: sound waves, electric current, air-fluid, liquid flow, etc. Waves can interfere with each other.Thus the interference of waves can be a clue to get pink noise.

Small system and seemingly long memory
It is bizarre that an ultra-low frequency signal can come from a very small system.As an extreme example 6 , the semiconductor films of 2.5nm layers give observable pink noise.A small semiconductor can have pink noise down to 10 −7 Hz 7 , and voltage fluctuations through a semiconductor show pink noise from about 1Hz to 10 −6.3 Hz 8 .These remarkable low frequencies sound almost impossible for ordinary small systems.In this context, if the Wiener-Khinchin theorem were correct, then the strong low-frequency signal in S(ω) of the pink noise would necessarily indicate the non-vanishing long-time correlation x(t)x(t − τ) .Therefore, the Wiener-Khinchin theorem may not hold for pink noise.

Apparent no lower cutoff in the PSD
It is often discussed that the pink noise does not seem to have an explicit lower cutoff in the PSD determined by any physics governing the system.Therefore, the system exhibiting pink noise may not be in a stationary state.Therefore, it may be useless to have discussions based on the stationarity of the system.

Independence from dissipation
It is remarkable that the pink noise appears even in the Hamiltonian mean-field (HMF) model, which is a strictly conservative system 9 and has nothing to do with dissipation.Thus the usual fluctuation-dissipation theorem of the type δ x2 ∝ RkT may not hold for the pink noise(R is the electric resistance and kT is the temperature).

Square of the original signal
When deriving the pink noise, it is often the case that the original time sequence is squared prior to the PSD analysis.For example, in the case of music 10 , the sound wave data should always be squared for PSD; the authors claim that this squared data is the loudness.Similarly, in the case of the HMF model 9 , the authors always take a square of the original variables in order to obtain the pink noise.In both cases, the original data before taking the square does not show any pink noise.In the case of the electric current, this procedure is not manifest, although the seminal paper 1 emphasizes the square of voltage V 2 for PSD.
From the above five clues, we speculate that the beets of many synchronized waves may be the origin of 1/f noise.A simple superposition of two waves sin(ωt + λ t) + sin(ωt − λ t) = 2 cos(λ t) sin(ωt) with ω ≫ λ > 0 has no low frequency component around λ in the PSD.On the other hand, the square of the superposed wave above has a low-frequency signal, i.e., the beats, around 2λ in its PSD.Incidentally, it is sometimes confusing that the wave beat is "audible" although the PSD of the original superposition of the two waves does not show the corresponding low-frequency signal.
The above argument reminds us of a typical musical instrument, the Theremin 11 , which uses the wave beat.By mixing the high-frequency signals of 1000kHz and 999.560kHz generated by an electric circuit, the low-frequency signal of 440Hz can be extracted as audible sound.The latter frequency can be varied slightly by the player's hand, antenna distance, and capacitance, to produce the desired frequency signal.Thus the amplitude modulation can produce arbitrary low-frequency signals within a small-size system.The modulated signal has no intrinsic memory and has nothing to do with dissipation.
Another familiar device is the AM radio which clearly shows the wave beat or amplitude modulation (AM).By using 526.5kHz to 1606.5kHz radio waves, the low-frequency audible signal is extracted.In this case, the rectification (demodulation) process is essential to obtain audible low-frequency signals.This demodulation process is also essential for the pink noise in our proposal.In later sections, we will see a variety of pink noises in the many ways to demodulate.
The above five points will also be an elementary verification of our proposal.This will be discussed in later sections.There appear to be several causes of the wave beat that forms pink noise, but the concentration of the wave frequencies is the essence of low-frequency signals.We will now focus on such causes separately in the following sections: a) cooperative waves, b) resonance, and c) infrared divergence.

Beats from cooperative waves
In this section, we will analyze the cause of wave beat, especially when the frequencies of the waves spontaneously approach with each other.We consider cooperative systems that exhibit this behavior.

Exponential Approach
The most typical type of synchronization would be the exponential approach, such as in the case of the Kuramoto model 12 , ω = e −λ t where ω is the frequency and λ is the approach speed, and t is the time.Then the frequency distribution function P(ω) and the time distribution function p(t) are related to each other by P(ω)dω = p(t)dt.If we assume the stationarity of the fluctuation, we set p(t) ≡ p = const.Then, ( It is interesting that the exponential function gives the power index exactly −1. The observed beat is the interference of the pair of frequency distributions above, and the beat frequency ∆ω has its probability distribution function Q(∆ω) as which again is proportional to (∆ω) −1 with small modification factor of ln[...∆ω].The detail of the full form Q(∆ω) depends on the boundaries of the integration domain ω 1 < ∆ω < ω 2 .Typical examples are shown in Fig. 1.The pink noise is robust, and the frequency distribution is directly reflected in the PDF of the waves at those frequencies, where ω is a fiducial frequency, c is a mixing constant, r i the Poisson random variable in some range for each sinusoid and i runs from 1 to some upper limit.This is demonstrated in Fig. 2 where the PSD of φ 2 is shown.The pink noise is robust, and the randomization of each phase of the sin-wave does not change the PDF except that the power index is slightly reduced, as shown in Fig. 3.
It is essential that the square of the signal φ 2 does show pink noise in PSD as in Fig. 1 while the original signal itself φ does not show any feature at low-frequency region as shown in Fig. 4.This fact manifestly demonstrates the pink noise comes from the wave beat.2, but the sine waves are superimposed with random phase θ i for each:sin (2πω(1 + ce −r i )t + θ i ).The power index drops a bit to −0.7, but this PDF shows the robustness of the pink noise from the wave beat.

Power Approach
Another popular type of synchronization would be the power approach ω = t −α .Repeating the same calculations as above, we obtain the frequency distribution function as where c ≡ pα −1 , β ≡ 1 + 1 α .The probability distribution function Q(∆w) of the beat frequency Q(∆w) is given by Then, if we expand with respect to small ω 1 and small ∆ω.The exponent is less than −1 for α > 0, and greater than −1 for α < 0 but the fiducial power is −1.Typical examples are shown in Fig. 5.
A typical wave signal can be constructed as before, and PSD for φ 2 are shown in Fig. 6 for α = 3, and in Fig. 7 for α = −3.
Although the above demonstrations are typical simple models of the cooperative waves, the frequencies are fixed.However, it is also possible to consider dynamical cooperative systems with time-dependent frequencies, and they often show pink noise; macroscopic coupled spin models 13 and the Hamiltonian mean field model 9 .Since the discussion of these is beyond the scope of this paper, we will cover them in a separate paper soon.

Beats from Resonance
We now consider the resonance, which produces the spontaneous concentration of frequencies and the wave beats.When the system with the intrinsic eigenfrequency Ω is stimulated (repeatedly), it emits the wave mode of the frequency Ω as well as  those close to Ω. Resonance thus ensures the concentration of frequencies in a small range.Since these frequencies are close to each other, the waves of these frequencies beat and produce a signal in low-frequency regions.
Suppose a typical case of the resonance characterized by the resonance curve, the Cauchy distribution where Ω is the resonance frequency and κ characterizes the sharpness of the resonance.We will interpret that this function R[ω] as proportional to the number of ω-modes in the resonator.Then the frequency distribution function P(ω) is given by the inverse function of R[ω], as where we have chosen the upper half of the inverse of R[ω], since the lower half is symmetric to the upper half.It is possible to make a naive approximation of Eq.9 by the exponential function ω = Ae −Bt , where the constants A, B are determined at the inflection point of Eq.9, as shown in Fig. 8.We already know that this exponential function gives the exact pink noise of slope −1 in PSD.This is demonstrated in Fig. 9, where the PSD is plotted for the square φ (t) 2 of the time sequence φ (t) generated by However, the system analysis is not easy.Using the relation P(ω)dω = p(t)dt with p(t) ≡ p = const, we obtain the frequency distribution function P(ω) as which cannot be reduced to a single power form if κ is finite.Further complications arise from the actual resonant system, which has complicated overtones and multiple eigenfrequencies that systematically contribute to the pink noise.A fully systematic derivation of pink noise for each concrete resonant system requires further investigation.Since this is beyond the scope of this paper, we do not discuss it further here, but it will be analyzed in a separate paper soon.

Beats from IR divergence
We now consider the third cause of the spontaneous concentration of frequencies from the infrared divergence.This class of systems exhibiting pink noise is quite diverse, but can be reduced to the system composed of electrons and photons described by electrodynamics.In this context, a quantum origin of pink noise was once proposed by using quantum interference 14,15 .It claims that the scattered electron state, after emission of a photon of frequency ω, and the unscattered electron state interfere with each other to produce a beat of frequency ω.However, this theory has been criticized 16,17 , mainly because quantum interference does not really occur; the scattered and non-scattered states are orthogonal to each other and have no chance to interfere.Even the introduction of the coherent state basis does not work.Incidentally, some other criticisms are not valid.
The essence of the pink noise is not the quantum interference, but the back-reaction of the emission of massless particles to the classical current and the classical wave beat interference.In this paper, we focus on such a classical description of electromagnetism.
In the semiconductor, the electric current can be classical beyond the scale of the free streaming length, about 10nm, which is several tends of the lattice size.When the system size is about 1mm, there are 10 10 classical current elements.When the electron meets any impurity, it changes its momentum from p i to p f , emitting the photon of momentum p i − p f .Starting with the classical current, the number of emitted photon is given by which is IR divergent 18 , and ε is the polarization vector.We assume that the average classical electric current has a fiducial frequency Ω, which is determined by the applied voltage and the conductivity before the interaction with the impurities.Each scattering with the impurities emits light of energy ω and exerts a back-reaction on the current of amount energy-shift ω with the probability 1/ω (bremsstrahlung).Then the original current cascades into the superposition of an enormous number of local currents with frequencies Ω − ω i , i = 1, 2, ...N.Then each pair of these currents makes beats with all the possible differences (Ω − ω i ) − (Ω − ω j ) = ω j − ω i , 1 ≦ i, j ≦ N.This process is the same as the previous case of the exponential approach 3.1, and many classical currents with slightly different frequencies interfere to give the wave beat as in Eq.( 2) and thus the pink noise as in Fig. 2.
In any case, quantum interference is not the essence of the origin of pink noise, but the classical synchronized waves are crucial.In this context, the coherent dressed state formalism for QED was developed to cancel the infrared divergence associated with the massless photon.This theory is well summarized in 19,20 , although most authors assume (semi-)classical background currents ab initio, and the classical degrees of freedom are not correctly derived.
The derivation of the classical degrees of freedom in QED is possible in the closed time-path formalism of the effective action associated with an unstable state.The IR divergence of the theory requires the separation of the classical statistical kernel from the complex effective action.Then the Langevin equation with classical noise is derived from the effective action and can describe the classical evolution of currents 21 .This formalism requires a more systematic discussion than we can give here.However, we will report this theory, including the classical-quantum interference, in a separate paper.

Discussions
So far, we have proposed three kinds of origin of the synchronizing waves, which gives systematic beats and produce pink noise.Since the pink noise is generated by the wave beat or the amplitude modulation, any demodulation process is required for observation.This demodulation process may be a)intrinsic mechanisms associated with the system or b) operational processes associated with the data reduction for PSD.In either case, the demodulation process provides robustness and a variety of pink noise.This section is devoted to showing some examples of such robustness and variety.
1. fiducial: The fiducial signal is the one discussed in 3.1, with the same parameters of Fig. 2: ω = 10, c = 0.2, and r i is a random field in the range [0,30].There, 10 3 sinusoids are superimposed according to Eq.7.The squared signal φ 2 shows a clear pink noise of slope −1.0 as in Fig. 2.
2. the threshold for φ 2 : We set the new data zero for the φ 2 data that is smaller than the mean and leave the other data as they are.The PSD shows pink noise with slope -1.0, almost no change from the fiducial case.This case may apply to the nerve system, where only a voltage greater than some threshold can produce a spike signal.
3. on-off threshold for φ 2 : We set the new data zero for the φ 2 data that is less than the mean and set the other data to 1.
The PSD shows pink noise with a slope of -0.94.
4. on-off inverse threshold for φ 2 : This is the opposite of case 3. We set the value 1 for the φ 2 data that is smaller than the mean and set the other φ 2 data to 0. The PSD shows pink noise with a slope of -0.94, exactly the same as in case 3.
5. threshold for original data φ : we set the new data zero for the φ data that is smaller than the mean and set the other data as is.The PSD shows pink noise with a slope of -0.98.
6. rectification of the original data φ : We set the new data to zero for the φ data that is negative and leave the other data as is.The PSD shows pink noise with a slope of -1.2.This may apply to some electric circuits contaning transistors, diodes, and vacuum tubes.
7. sequence of locally averaged φ 2 : We divide the entire time sequence of φ into 10 3 segments and apply a quadratic average in each segment.The PSD shows pink noise with a slope of -1.1.This is the data treatment in the original experiment 1 .
8. sequence of locally averaged φ : Same as case 7, but we apply a simple average in each segment.The PSD shows NO pink noise at all, and the power is positive +0.8. 9. coarse time resolution for φ 2 : We reduce the number of sample points to half of the original.The PSD shows an almost pink noise with a slope of -1.1.
10. fewer superimposed waves: We reduce the number of superimposed waves from the fiducial 10 3 to 10.The PSD shows NO pink noise.
11. more superimposed waves: We increase the number of superimposed waves from the fiducial 10 3 to 10 4 .The PSD shows pink noise with a power of -0.94.
12. longer time sequence: We extend the time sequence from the fiducial 10 4 to 10 5 .The PSD shows pink noise with a slope of -1.0; the same as before, but with a power law extended by a decade.
13. multiple fiducial frequencies; We changed the fiducial frequency from the original single to 5, randomly selected from 0 to 20.The PSD shows pink noise with a slope of -1.5.
As examined above, there are multiple demodulation processes.They are classified as a) system-intrinsic and b) operational in the data reduction, although the classification is not exclusive.Examples of a) are thresholding and rectification: cases 3,4,5,6.

9/12 7 Conclusions and prospects
We have discussed the origin of pink noise from the beat of cooperative waves.We have examined three possible causes for this cooperative effect: synchronization, resonance, and IR divergence.There may be more mechanisms.We point out the verifiability/falsifiability of our model based on the five crucial observations for the pink noise in section 2.

Wave
The wave is essential for producing beat and amplitude modulation.The wave may be hidden inside the system, and the data may be obtained after it passes through the threshold.If we cannot find a coherent wave in the system, our model cannot be applied.

Small system and apparent long memory
Although the amplitude-modulated fluctuation, the primary fluctuation, may accept the Wiener-Khinchin theorem, the demodulated fluctuation, the secondary fluctuation, does not accept the theorem because the secondary fluctuation does not appear in the PSD before any demodulation process.If we find the successful Wiener-Khinchin theorem for pink noise, our model cannot be applied.

Apparent no lower cutoff in the PSD
The beat of the cooperative wave or the amplitude modulation can yield an infinitely low-frequency signal from inside a finite system within the observational constraints.Therefore, if an intrinsic lower-cutoff frequency is found in the pink noise, our model cannot be applied.

Independence from dissipation
The beat of the cooperative wave or the amplitude modulation is a secondary fluctuation caused by wave synthesis.Therefore, the dissipation may destroy the pink noise because it may cancel the fragil wave beats.

Square of the original signal (necessity of the demodulation process)
The amplitude modulation needs some demodulation process for observation.The primary fluctuations before the demodulation do not appear in the PSD.Our model for pink noise predicts the demodulation process as either a)intrinsic to the system or b)operational in the data reduction.If the demodulation is found in the system of pink noise, and the pink noise disappears when the demodulation process is removed, then our model is strongly favored.
Although we have proposed a basic model of pink noise, we still have many problems with elaborating the present formalism.Some of them have already been described in appropriate places with the keyword 'separate paper'.They are dynamical cooperative systems, actual resonant systems, and systems with IR divergence.Among them, we summarize the possibly resonant systems in Table 1.
The list in Table 1 is tentative and incomplete.It will be completed in our future publications, including the verification of our simple pink noise model.
acknowledgments We would like to acknowledge many valuable discussions with the members of the Lunch-Time Remote Discussion Meeting, with the members of the Department of Physics Ochanomizu University, with Manaya Matsui and Izumi Uesaka at Kyoto-Sangyo University.

Figure 2 .
Figure 2. The PSD of φ 2 is shown with ω = 10, c = 0.2, and r is a random field in the range [0,30].1000 sine waves are superimposed according to Eq.3 The power index can change up to about 0.1 for each run.This PDF shows the pink noise of index -1 for four decades.

Figure 3 .
Figure 3. Same as Fig.2, but the sine waves are superimposed with random phase θ i for each:sin (2πω(1 + ce −r i )t + θ i ).The power index drops a bit to −0.7, but this PDF shows the robustness of the pink noise from the wave beat.

Figure 4 .
Figure 4. Same as Fig.2, but this is PDF for the original signal φ .Pink noise never appears in this case, indicating that the noise arises from the wave beat.

Figure 6 .
Figure 6.The PSD is shown for φ 2 with α = 3, ω = 440, c = 0.3, and r i is a random field in the range [0,20].200 sine waves are superimposed according to Eq.7 This PDF shows the pink noise of index -1.3 for four decades.

Figure 7 . 12 Figure 8 .
Figure 7.The PSD is shown for φ 2 with α = −3, ω = 440, c = 0.01, and r i is a random field in the range [0,1].200 sine waves are superimposed according to Eq.7 This PDF shows the pink noise of index -1 for three decades.

Figure 9 .
Figure 9. PSD of the time sequence φ (t) 2 generated by Eq.(10) with κ = 0.1, Ω = 10, and the domain of the random field r i is [0, 10].We have superimposed 100 sine waves, and this PSD shows the approximate power law of index −1.2.

Table 1 .
List of systems that show pink noise possibly due to the resonance effect, as we have discussed in 4.This Table is preliminary, and the final analysis will be reported in our papers soon.