1/f spectral trend and frequency power law of lossy media

Transcription

1 1/f spectral trend and frequenc power law of loss media W. Chen Simula Research Laborator, P. O. Box. 134, 135 Lsaker, Norwa (5 Ma 3) The dissipation of acoustic wave propagation has long been found to obe an empirical power function of frequenc, whose exponent parameter varies through different media. This note aims to unveil the inherent relationship between this dissipative frequenc power law and 1/f spectral trend. Accordingl, the 1/f spectral trend can phsicall be interpreted via the media dissipation mechanism, so does the so-called infrared catastrophe of 1/f spectral trend 4. On the other hand, the dissipative frequenc power law has recentl been modeled in time-space domain successfull via the fractional calculus and is also found to underlie the Lev distribution of media, while the 1/f spectral trend is known to have simple relationship with the fractal. As a result, it is straightforward to correlate 1/f spectral trend, fractal, Lev statistics, fractional calculus, and dissipative power law. All these mathematical methodologies simpl reflect the essence of complex phenomena in different fashion. We also discuss some perplexing issues arising from this stud. Kewords: frequenc power law dissipation, 1/f spectral trend, fractal, Lev statistics process, fractional calculus, Hurst exponent, correlation function, complex phenomena 1. Power law dissipation and 1/f spectral trend The dissipation of acoustic wave propagation through loss media is generall described b I α ( f )t = I e, where I represents the energ of an acoustic field variable (signal) such as velocit or pressure, t is the traveling time, and f frequenc. The attenuation coefficient α(f) is experimentall found to obe a frequenc power law function 1-3

2 ( ) α f α =, [,], (1) f where α and are empirical media-dependent parameters. For most solid and highl viscous materials, is close to, while for most of human tissues, ranges from 1 to 1.7. The underwater sediments and rock laers have 1, and for boundar laer loss of rigid tubes, =.5. The power spectral (frequenc) P of a dissipative acoustic signal can be calculated b P α f t ( f ) = + I e dt = I α f. () Note that the unit of power P is the square of the signal unit rather than necessaril phsical power (energ). It is seen from () that the frequenc power law (1) of dissipative acoustic media underlies 1/f spectral trend, which means the power spectral of a signal is inversel proportional to the frequenc f according to a 1/f β power law 4-6 (β=, P 1 f ). Here β is a nonnegative real number. β=1 is found in most phenomena, which causes the terminolog 1/f spectral trend. It is noted that a singularit develops at β=1. For =1 dissipation, we also encounter the singularit issue in its corresponding fractional derivative model equation 7, and man biomaterials have the exponent around 1 (newl termed It is also known 4 that the turbulent velocit of large Renolds numbers follows the spectral trend 1/f 5/3. Correspondingl, =5/3 in terms of (1).. The infrared catastrophe For the low frequenc power P( f ) df = I α f df, 1 leads to infinite power (divergent), colorfull termed the infrared catastrophe 4. The frequenc power law of dissipation is found to take effect over a finite range of frequenc, namel, p f f f min max p, which echoes the self-similarit extends usuall onl over a finite range in real phsical problems 8. In terms of the power law dissipation, the 1/f

3 power spectra do not hold for zero frequenc component of a signal and avoids the infrared catastrophe. On the other hand, the attenuation parameter expression ( ) α + f = α 1 α f, [,], also fits well with measured data of man dissipative acoustic signals 9, where α 1 is also an empirical parameter. Accordingl, we have Pˆ I α + α f α1 α f t ( f ) = + I e dt = 1. (3) Here the 1/f power spectral is revised as 1/(α 1 +α f β ). It is obvious that (3) circumvents the infrared catastrophe. It is also interesting to connect the power law dissipation with the Wiener-Kinchin spectrum function in terms of page 18-4 of Mandelbrot [4]. We can accordingl assume ( f ) α f Q( ) α =, which varies ver slowl when f close to. f 3. Hurst exponent The Hurst exponent is an essential measure of the smoothness of fractal time series. It is well known that the exponent parameter β of 1/f signal has a simple relation with the Hurst exponent H where H [,1]. Thus, we have β=h+1, (4) =H+1. (5) However, the analsis shows that the relationship formulas are not correct or at best onl hold for a limit range of the value of H. For example, β== underlies a Gaussian process, and the corresponding H=.5 indicates the Gaussian Brownian motion. The formula (4) is right here. But for =1, the formula (5) produces H=, corresponding to the

4 white noise 6. However, it is known that =1 dissipation underlies the Cauch statistic distribution 7 and, under certain condition, it is a deterministic process rather than the white noise. =1 should correspond to H=1 for a deterministic process rather then H=. In fact, the relationship formula (4) has been controversial when H= and β=1 for ears. β=, corresponding to H=-1/ via (4), is often considered the white noise in terms of the 1/f spectral trend. Therefore, the relationship formula (4) and (5) do not hold. 4. Fractional calculus model, correlation function, and fractal dimension The linear wave equation model of frequenc-dependent dissipation, developed b Chen and Holm [7], is stated as 1 p = c p α + 1 c ( ) p, (6) where p is the pressure signal, and ( ) is the fractional Laplacian of the order. The second right-hand term is to describe the dissipation of arbitrar frequenc dependenc, whose order is the value of as in the frequenc power law (1). According to the equivalence of exponent parameters of the 1/f spectral trend and the power law dissipation, one can find the simple link between the fractional calculus model and the 1/f spectral trend. B using both the fractional space/time derivatives, we have the fractional diffusion wave equation 1,11 µ p s = γ ( ) p µ s, 1 µ, = s µ + 1, (7) where γ is the viscous constant, s and µ are real number. The Fourier analsis 1 shows that (7) satisfies the power law (1). For µ=s=, equation (7) is the normal wave equation u = γ u with =; for µ=1, s=, it is the normal diffusion equation u = γ u

5 with =; for non-integer µ and s, it is anomalous diffusion equation with =s-µ+1 (not for s=µ=1,) and essentiall accounts for non-local and memor effects (entrop) on energ dissipations underling a random walk (fractional Brownian motion) 13. More precisel, the closer is to 1, the longer period dependenc (stronger memor). Let s=, we have the fractional Signal equation µ p µ = γ p. (8) In terms of (8), it is noted that µ=h, where H is the Hurst exponent, seems reasonable. For µ=h=, the equilibrium equation p = p underlies the frequenc (time) independent white noise; for µ=1 and H=.5, the normal diffusion equation u = γ u reflects the Gaussian process; for µ= and H=1, the normal wave equation u = γ u describes the non-dissipative deterministic wave propagation process. However, <µ<1, <H<.5 corresponds to <<3. This excessive dependenc of dissipation on frequenc is rarel, if ever, found in the real world and contradicts that = underlies the frequenc independent dissipation and white noise signal. This puzzle perplexes me deepl. Equation (7) is also closel related to the autocorrelation function through x t µ s ( t τ ) corr( τ ) dτ t =, (9) where corr(τ) is the velocit autocorrelation function, and x the position variance of random variable (mean square deviation). <µ<1 and s= indicates the sub-diffusion, while µ>1, s= or µ=1, <s< means the super-diffusion process 1. Is H=µ/s? Let µ=1, we have the anomalous diffusion equation p = γ ( ) s p. (1) Observing (1), we find that s [1,] appears compatible with H [1,.5], s [,1) compares well with H [,.5). There is the singular jump around s=1. Note that the

6 exponent in (1) equals s in terms of (1). For most media, varies from 1 to, correspondingl, the majorit of signals have H from 1 to.5. It is well known that the Hurst exponent is related to the fractal dimension D b D = d + 1 H, (11) where d is the topological dimension. The frequenc dissipative power law (1) can be restated as ( f ) lnα α =. (1) ln f (1) underlies a self-similarit with the fractal dimension, called the dissipative dimension 7. In terms of the connection between the 1/f spectral trend and multifractal 4, we assume that the dissipative dimension ma collectivel underlies the multifractal. The Gaussian dissipative dimension is which echoes the second definition of the fractal dimension in page of Mandelbrot [4]. Including the topological dimension, the fractal dimension is D =d+-, (How about D =d+1-/? No). 5. Lev stable distribution In the context of kinetic phsics, the frequenc dissipative power law (1) also underlies the Lev -stable statistic distribution 7, whose probabilit densit function is the fundamental solution of the corresponding anomalous diffusion equation (1), where s=. [,] coincides the fact that the Lev stable index ranges from to. Accordingl, we have β [,] of the spectral trend for dissipative acoustic signals in terms of the Lev stable distribution. In terms of (1), the growth of the position variance can also be evaluated b the spatial Lev stable distribution function S (x,t)

7 x ( x, t) dx t = x S. (13) 6. Remarks Despite some perplexing issues, we have b far connect the 1/f power spectral, Hurst exponent, fractional calculus equation, fractal, and Lev stable process through the frequenc power law of dissipative media. Let us summarize some definite results (1 dimension): 1) for a Gaussian process, D=1.5, H=.5, β==; ) for a white noise process, D=, H=, β==; 3); for a deterministic process, D=1, H=1, β==1. The 1/f power spectral is observed in a broad variet of phsical, chemical, and biological phenomena. The present analsis shows that the frequenc power law dissipation (1) is useful to describe all these 1/f power spectral signals. The revelation of these underling relationships has some potential applications. For instance, in the solution of inverse problem, we can first evaluate the exponent β of the 1/f signal, and then have =β. The signal noise distribution obes the Lev -stable distribution, the signal noise can be removed via 1/f β filter, and the known parameter can directl be used to construct the corresponding linear or nonlinear dissipative wave equation model 7. Appendices: The power spectral can also be evaluated b the squared modulus of the Fourier α t ibft transform, i.e., FT { I e } I ( f bf j ) [( f j ) b f ] f + = α + + ω α + ω +, where ω is the angular frequenc. It is noted H@.75, and β@1 (@ means ) have been observed in man cases. Is there a possible correspondence between H [.5,1] and [,], where H=.75,1 respectivel corresponds to =1,? In addition, ma we assume =1/H for 1 and =H/ for <1 in terms of the fractal? Here we also dare to assume that the averaged dissipation of the high Renolds number turbulence follows the Lev -stable distribution.

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