Fourier spectrum

Figure 5.51. STM images (unfiltered) and corresponding Fourier spectra of the (a) sodium-cleaned and (b) sodium-dosed Pt(l 1 l)-(2x2) 0 adlattice. Total scan size 638 A, other conditions as in Figure 5.50.78 Reprinted with permission from Elsevier Science.

Figure 18. (A) Fourier transformation spectra of the time trace of surface pressure for the steady loop (see Figure 17). Top real part (elastic component), bottom imaginary part (viscous component). (B) Inverted Fourier Spectra for the real and imaginary parts.

Figure 18A shows the Fourier spectra thus obtained. The real and imaginary parts correspond to the elastic and viscous components of the DOPC thin film, respectively. We can see that the spectrum is composed not only from the fundamental (coo) but also from the higher (2harmonic components. Such a trend indicates that the DOPC thin film exhibits rather large nonlinearity in the viscoelastic characteristics. 

FIGURE 7.11 Time-resolved single-beam CARS transients (left column) and corresponding Fourier spectra (right column) for bromotrichloromethane (CBrClj), chloroform (CHClj), bromoform (CHBrj) and a mixture of the three constituents. This result shows that the different components can clearly be distinguished by their characteristic vibrational resonances, (von Vacano and Motzkus 2007b). (From von Vacano and Motzkus, Phys. Chem. Chem. Phys., 10 681-691, 2008. Used with permission.)... 

Fourier transform for different, chemically very similar halomethanes and a mixture thereof. The time-domain data in Figure 7.11 can be directly interpreted as an observation of molecular motion in real time, made possible by the compressed ultrashort pulses in the microscope. From the presence of different oscillatory patterns and beatings, it already becomes clear that the different molecules can be discriminated with high resolution. Correspondingly, the Fourier spectra in Figure 7.11 show markedly different vibrational resonances, which can also be discriminated in the ternary mixture of all components. 

As can be seen from the experimentally measured Fourier spectra of the time-resolved vibrations (Figure 7.12), shaping of the excitation pnlse into a mnltipnlse sequence with a temporal separation b (Figure 7.10d) allows the selective preparation... 

Although the present work is concerned primarily with spectra, its applicability does not lie only in that area. The term spectra should be understood to include one-dimensional data from experiments that do not explicitly involve optical phenomena. Data from fields as diverse as radio astronomy, statistics, separation science, and communications are suitable candidates for treatment by the methods described here. Confusion arises when we discuss Fourier transforms of these quantities, which may also be called spectra. To avoid this confusion, we adopt the convention of referring to the latter spectra as Fourier spectra. When this term is used without the qualifier, the data space (nontransformed regime) is intended. 

Extrapolation of discrete Fourier spectra is accomplished by applying the constraints to the discrete Fourier series given by the inverse DFT. 

The Fourier spectra of concentrations and of the reaction rate are quite similar. The difference is that the strong concentration decay suppresses the Fourier amplitudes at high frequencies. For the concentration motion the change in time of the K (t) acts as an external noise from which the concentration motion selects the main frequencies. 

Figure 14.4 Phase portrait (dotted experiment, solid simulation) and experimentally determined and calculated Fourier spectra of the response function with the excitation of the amplitude 14.6 V...

By increasing the amplitude of the excitation (16,4 V) a period doubling can be observed. Figure 14.5 shows the phase portrait and the Fourier spectra of the response function in the... 

Figure 14.9 Phase portraits of the resonance circuit with a TGS-crystal at different driving voltages below the phase transition and the corresponding Fourier spectra of the response functions with the periods T, 2T, 4T and deterministic chaos...

A thorough analysis of chaotic oscillations for the NH3/O2 reaction over Pt has been performed by Sheintuch and Schmidt (2//). Bifurcation diagrams were presented in great detail, as well as phase plane maps and Fourier spectra. Figure 18 shows a series of oscillation traces obtained for various oxygen concentrations. By extracting a next return map from trace g in Fig. 18, evidence for intermittency could be obtained. 

Figure 5. Left Classical data for the force-force correlation function. Middle Fourier spectra for the correlation function. Right The corresponding coarse-grained Fourier spectra. The lifetime width parameter y = 3 cm-1.

VMRIA VMRIA a program for a full profile analysis of powder multiphase, neutron diffraction time of ffight (direct and Fourier) spectra, V. B. Zlokazov, V. V. Chernyshev, J. Appl. Crystallogr., 1992, 25, 447 451 and DELPHI based visual object oriented programming for the analysis of experimental data in low energy physics, V. B. Zlokazov, Nucl. Instrum. Methods Rhys. Res. A, 2003, 502(2 3), 723 724 Rietveld refinement of TOF data... 

Figure 19.3 shows typical traces of time-resolved SHG from alkali-covered Pt(lll). In both cases, clear oscillatory components appear and they are ascribed to nuclear wavepacket motion of surface modes. There exist more than two components, which becomes clear by Fourier transforming the time-domain data (Figure 19.4). The Fourier spectra are obtained from the raw data with a delay time larger than 50 fs by subtracting background components whose frequencies are less than ITHz. For Cs adsorbate, a peak at 2.3 THz is prominent and is due to the Cs—Pt stretching mode, while the corresponding stretching mode is observed at 4.8 THz for K adsorbate.

Figure 19.4 Fourier spectra ofthe oscillatory components in time-resolved SHG traces from Pt(l 11) covered with Cs (a) and K (b).

A similar enhancement of the dephasing rate has been observed for K/Pt(lll). Figure 19.7 shows Fourier spectra of the oscillatory components when varying the pump fluence. It is evident that K—Pt stretching mode at 4.8 THz shows a marked red shift and broadening. This feature can be ascribed to the incoherent excitation of the lateral modes, as in the case of Cs/Pt(lll), but also notable in this system is that... 

Figure 19.9 Fourier-transformed spectra of the oscillatory parts of time-resolved SHC traces obtained by varying the repetition rate (solid curves). Trace (a) is obtained by single pulse excitation. The repetition rate was tuned to (b) 2.0, (c) 2.3, (d) 2.6, and (e) 2.9THz. The Fourier spectra of the excitation pulse trains are shown with dashed curves for each case. [38].

Fig. 4.4.4 [Bar2] Morlet wavelets (top) and their Fourier spectra (bottom) according to eqns. (4.4.21 a and b). Time and frequency are scaled in arbitrary units. Dilatation parameter from left to right a = 0.5,1.0,2.0 Widths of WabO) = 3,6,12 (top). Widths of Wai,((u) = 3.0,1.5,0.75 and corresponding peak positions at 12,6,3 (bottom).

The Morlet wavelet can be understood to be a linear bandpass filter, centred at frequency m = coo/a with a width of /(aoa). Some Morlet wavelets and their Fourier spectra are illustrated in Fig. 4.4.4. The translation parameter b has been chosen for the wavelet to be centred at time f = 0 (top). With increasing dilatation parameter a the wavelet covers larger durations in time (top), and the centre frequency of the filter and the filter bandwidths become smaller (bottom). Thus depending on the dilatation parameter different widths of the spectrum are preserved in the wavelet transform while other signals in other spectral regions are suppressed. 

Figure 14. Fourier spectra of the residuals of the decays of Fig. 13. Bands in the middle spectrum are labeled with values in GHz. Although there appear to be more than three components in the lower spectrum, only the ones at 1.0,9.7, and 10.7 GHz are reproducible.

Figure 42. Fourier spectra of the decays of Fig. 41. The peaks are negative because the 1 GHz component in the decays has a — 1 phase.

Figure 15.6 (a) Fourier spectra ofthe oscillatory absorbance changes of Fig. 15.5 (d) - (f). The spectra are scaled relative to each other and displays low-frequency modes, (b) Low-frequency spontaneous Raman spectrum of acetic acid (taken from Ref [35]). 

In this section we illustrate the characterizations which can be given to the Fourier spectra of chaotic solutions of quasi-integrable systems. We first consider the easy integrable case ... 

Figure 15. Fourier spectra (52) computed for asteroid 305 Gordonia (lower one computed using a 100 Myr integration, upper one using a 10 Myr integration with more frequent sampling).

Another technique makes use of olefins to spin-tixq) short-lived Mu and make it observable in muonated fi ee radicals. In this way it was possible to prove that Mu is formed also in liquid chloroform or acetonitrile where it was never directly detected. This is shown in Figure 4 which presents Fourier spectra of the FID that was obtained with 0.2 M solutions of 2,3-dimethylbutadiene-l,3 (DMBD) in different solvents. At this concentration, DMBD can trap Mu on a time scale of ca. 40 ps in CHCI3 [17], The spin trap adduct is the 1,1,2-trimethylallyl radical which has the Mu atom substituted in the exo-methyl group. It shows up in the spectrum by two muon precession fi equencies, denoted R, which are the equivalent of the electron-nuclear double resonance (ENDOR) transitions in conventional magnetic resonance of fi ee radicals. 

The OPT occurs at p = 1 via a subcritical Hopf-type bifurcation where the system settles to a uniform precession state with a small reorientation amplitude (A 7T so that 0 system switches back to the unperturbed state at p = p] 0.88 where a saddle-node bifurcation occurs. The trajectory in the ux, Uy) plane is a circle whereas, in a coordinate system that rotates with frequency /o around the z axis, it is a fixed point. The time Fourier spectra of the director n have one fundamental frequency /o, whereas 0 , 4> and A do not depend on time. 

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