Power-spectrum

Many of the conventional measures used in studying dynamical systems - power spectra, entropy, Lyapunov exponents, etc. - can in fact be used to quantify the difference among Regimes I-IV ([kaneko89a], [kaneko89c]). 

One of the most common ways of characterizing complexity is by taking Fourier transforms. The spatial power spectrum of a time series of Ti, for example, is defined by 

We can similarly consider the behavior of the temporal power spectrum of a pattern, P(w), defined by 

Regimes I-III are each characterized by peaks immersed in broad-band noise. In regime IV the peaks are replaced by a Lorentzian band noise around [kaneko89]. 

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Finding the values of G allows the determination of the frequency-domain spectrum. The power-spectrum function, which may be closely approximated by a constant times the square of G f), is used to determine the amount of power in each frequency spectrum component. The function that results is a positive real quantity and has units of volts squared. From the power spectra, broadband noise may be attenuated so that primary spectral components may be identified. This attenuation is done by a digital process of ensemble averaging, which is a point-by-point average of a squared-spectra set. 

Chapter 6 is a short primer on CA and language theory, and provides a basic discussion of formal language theory, the relationship between CA and formal language theory, power spectra of regular languages and reversible computation. 

We give both forms here because, depending on the problem, either of the two forms may yield a simpler answer. Although the discrete form may appear a-priori to be the natural choice, for example, its discrete nature actually introduces some spurious periodicity into the resulting power spectra, thus washing away some of its characteristic features. We therefore follow Li [H87] in working mainly with the continuous forms of R t) and... 

Figures 6.4 shows some of the variety of possible shapes of P f) for elementary rules shown in the figures are the power spectra for rules Rll, R56, R150 and R200. The plots were generated for lattice size N = 2048, ignoring the first 15 transient steps and averaging a total of 20 runs. Also, since there are only N data points but 2N real Fourier components, half of the components are redundant. Thus, only the first half of the components are shown (see [H89b] or [H87] for a complete set of power spectra).

Li s algorithm for generating closed form solutions for the power spectra of CA attractors consists essentially of five main parts ([li87], [li89b]) ... 

Regime Spatial Power Spectra S(k) Pattern Distribution Q(.D) Pattern Entropy Sp Dynamical Entropy Sd... 

Much of the regularity in classical systems can often be best discerned directly by observing their spatial power spectra (see section 6.3). We recall that in the simplest cases, the spectra consist of few isolated discrete peaks in more complex chaotic evolutions, we might get white noise patterns (such as for elementary additive rules). A discrete fourier transform (/ ) of a typical quantum state is defined in the most straightforward manner ... 

The power spectra S(f) for transport phenomena in many diverse physical systems including transistors, superconductors, highway traffic and river flow ([bak88a],[carl90]) - has been experimentally observed to diverge at low frequencies with a power law f, with 0.8 < (3 < 1.4, Moreover, S f) obeys this power-law behavior over very large time scales. Commonly referred to as the l//-noise (or Bicker-noise noise) problem, there is currently no general theory that adequately explains the ubiquitous nature of 1/f noise. 

H87] Li, W., Power spectra of regular languages and cellular automata . Complex Systems 1 (1987), 107-130. 

Figure 2a. Profiles of the power-spectra of the true brightness distribution, the noiseless blurred image, and the actual data (noisy and blurred image). Clearly the noise dominates after frequency 80 frequels.

FIGURE 20.10 (a,b) Phase images of cryo-ultramicrotomed surfaces of triblock copolymer styrene and ethylene-butylene (SEES) samples of neat material and loaded with oil (40 wt%), respectively. (c,d) Phase images of film of triblock copolymer poly(methyl methacrylate-polyisobutylene-poly(methyl methacrylate) (PMMA-PIB-PMMA) immediately after spin-casting and after 3 h annealing at 100°C, respectively. Inserts in the top left and right comers of the images show power spectra with the value stmctural parameter of microphase separation. 

The following criteria are usually applied when analyzing these power spectra the magnitude of power, the frequency at which the power begins to fall off, and the slope of the descending part of the plot. From such an analysis one can sufficiently well identify the various types of corrosive attack (uniform, crevice, pitting). 

Figure 10.3 Dependence of the gradient of the characteristic length scales on the irradiation intensity observed for PSAF/MMA mixtures with different compositions at room temperature. The 2D power spectra corresponding to the morphologies are indicated in the inset.

Figure 7. TEM images of Ag-core/Au-shell particles in molar ratio Ag Au of (a, b) 1 1 (c, d) 1 3. Note the enlargement of the particles as the Au-shell becomes thicker (panels (a) and (c)) as well as the resolution of the lattice planes (panels (b) and (d)). The power spectra indicate that Au grows epitaxially onto the Ag-seeds. (Reprinted from Ref. [153], 2000, with permission from American Chemical Society.)...

The molecular collective behavior of surfactant molecules has been analyzed using the time courses of capillary wave frequency after injection of surfactant aqueous solution onto the liquid-liquid interface [5,8]. Typical power spectra for capillary waves excited at the water-nitrobenzene interface are shown in Fig. 3 (a) without CTAB (cetyltrimethy-lammonium bromide) molecules, and (b) 10 s after the injection of CTAB solution to the water phase [5]. The peak appearing around 10-13 kHz represents the beat frequency, i.e., the capillary wave frequency. The peak of the capillary wave frequency shifts from 12.5 to 10.0kHz on the injection of CTAB solution. This is due to the decrease in interfacial tension caused by the increased number density of surfactant molecules at the interface. Time courses of capillary wave frequency after the injection of different CTAB concentrations into the aqueous phase are reproduced in Fig. 4. An anomalous temporary decrease in capillary wave frequency is observed when the CTAB solution beyond the CMC (critical micelle concentration) was injected. The capillary wave frequency decreases rapidly on injection, and after attaining its minimum value, it increases... 

FIG. 3 Power spectra for capillary waves excited at the water-nitrobenzene interface (a) without CTAB molecules and (b) 10s after injection of a CTAB solution (0.5mL, lOmM) into the water phase. 

The effects of cumulative intravenous PCP on (A) the direct cortical EEG recorded in freely moving rats, and (B) the associated sequential l-minute EEG power spectra... 

Young, G.A. Neistada, L. and Khazan, N. Differential neuro-pharmacological effects of mu, kappa, and sigma opioid agonists on cortical EEG power spectra in the rat. Res Commun Psvchol Psvchiatr Behav 6 365-377, 1981. 

The applicability of these techniques depends on the type and form of the power spectra of both signal and noise. The only method that can be universally applied, is signal averaging. If the signal function is measured n times, the S/N ratio increases by Jn. 

Fig. 3.7. Power spectra of the most important types of noise (schematic representation in the frequency domain)... 

Kantor, S., Jakus, R., Bodizs, R., Halasz, P. Bagdy, G. (2002). Acute and long-term effects of the 5-HT2 receptor antagonist ritanserin on EEG power spectra, motor acivity, and sleep changes at the light-dark phase shift. Brain Res. 943, 105-11. 

Figure 29. Comparison of dimensionless power spectra of differential pressure fluctuations. Double probe across levels 2 and 3 x/L = 0.0, coal burning bubbling bed combustor. Full set of scaling laws with iron grit in cold bed hot bed material in cold bed violates scaling laws. (From Nicastro and Glicksman, 1984.)...

We have made Fourier transforms [40] of the data in figures 12 and 13, to deduce the power spectra of periodicities. The results are shown in figures 14 and 15. The same periods are found in deuterium and oxygen in the Japanese cedar, within experimental error, as indicated from the widths of the peaks of the power spectra, figure 14 they are listed in Table 1. 

Blackman, R. B., Tukey, S. W., The Measurement of Power Spectra, Dover, NY, 1958 and see also Weast, Handbook of Chemistry and Physics, 54th Edition, Chemical Rubber Company, Cleveland, OH, 1973. 

Miller, P. L. and P. E. Dimotakis (1991). Reynolds number dependence of scalar fluctuations in a high Schmidt number turbulent jet. Physics ofFluids A FluidDynamics 3,1156-1163. (1996). Measurements of scalar power spectra in high Schmidt number turbulent jets. 

The superionic phase has been explored with more extensive CPMD simulations.69 Calculated power spectra (i.e., the vibrational density of states or VDOS) have been compared with measured experimental Raman spectra68 at pressures up to 55 GPa and temperatures of 1500 K. The agreement between theory and experiment was very good. In particular, weakening and broadening of the OH stretch mode at 55 GPa was found both theoretically and experimentally. 

Dumont, R. S., and Brumer, P. (1988), Characteristics of Power Spectra for Regular and Chaotic Systems,/. Chem. Phys. 88, 1481. 

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