Patterns time-domain

Free induction decay A decay time-domain beat pattern obtained when the nuclear spin system is subjected to a radiofrequency pulse and then allowed to precess in the absence of Rf fields. 

The analogue between the Young- interferometer (interference pattern in the spatial domain) and the serrodyne modulated MZI (interference pattern in the time domain) is striking. An optimized Young interferometer17 can also show resolutions down to <3n 10 8. 

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. 

Fourier transform NMR spectroscopy, on the other hand, permits rapid scanning of the sample so that the NMR spectrum can be obtained within a few seconds. FT-NMR experiments are performed by subjecting the sample to a very intense, broad-band, Hl pulse that causes all of the examined nuclei to undergo transitions. As the excited nuclei relax to their equilibrium state, their relaxation-decay pattern is recorded. A Fourier transform is performed upon this relaxation-decay pattern to provide the NMR spectra. The relaxation-decay pattern, which is in the time domain, is transformed into the typical NMR spectrum, the frequency domain. The time required to apply the Hl pulse, allow the nuclei to return to equilibrium, and have the computer perform the Fourier transforms on the relaxation-decay pattern often is only a few seconds. Thus, compared to a CW NMR experiment, the time can be reduced by a factor of 1000-fold or more by using the FT-NMR technique. 

Figure 1.3 From the masking pattern it can be seen that the excitation produced by a click is smeared out in the time domain.

Thus, in order to simulate a perceptually convincing room reverberation, it is necessary to simulate both the pattern of early echoes, with particular concern for lateral echoes, and the late energy decay relief. The latter can be parameterized as the frequency response envelope and the reverberation time, both of which are functions of frequency. The challenge is to design an artificial reverberator which has sufficient echo density in the time domain, sufficient density of maxima in the frequency domain, and a natural colorless timbre. 

FT/ICR experiments have conventionally been carried out with pulsed or frequency-sweep excitation. Because the cyclotron experiment connects mass to frequency, one can construct ("tailor") any desired frequency-domain excitation pattern by computing its inverse Fourier transform for use as a time-domain waveform. Even better results are obtained when phase-modulation and time-domain apodization are used. Applications include dynamic range extension via multiple-ion ejection, high-resolution MS/MS, multiple-ion simultaneous monitoring, and flatter excitation power (for isotope-ratio measurements). 

Figure 4.2b is a presentation of the FID of the decoupled 13C NMR spectrum of cholesterol. Figure 4.2c is an expanded, small section of the FID from Figure 4.2b. The complex FID is the result of a number of overlapping sine-waves and interfering (beat) patterns. A series of repetitive pulses, signal acquisitions, and relaxation delays builds the signal. Fourier transform by the computer converts the accumulated FID (a time domain spectrum) to the decoupled, frequency-domain spectrum of cholesterol (at 150.9 MHz in CDC13). See Figure 4.1b.

Actually, the spectroscopic data would more closely resemble the pattern in Figure 3.15, which is the same as the wave in Figure 3.14, except that the overall intensity of the signal decays exponentially with time. (Note that the decay does not affect the frequencies.) Such a pattern is called the modulated free induction decay (FID) signal (or time-domain spectrum). The decay is the result of spin-spin relaxation (Section 2.3.2), which reduces the net magnetization in the, y plane. The envelope (see Section 3.6.2) of the damped wave is described by an exponential decay function whose decay time is T, the effective spin-spin relaxation time. 

Figure 1. Muonium spin precession signal for a ZnO powder sample at 5 K. The upper plot is the raw time-domain spectrum (corrected for the muon decay) while the lower plot is the corresponding frequency spectrum. The central line corresponds to the Larmor frequency of the bare muon (ionized muonium) and the two symmetrically disposed satelhtes are associated with muonium. The dotted curve is a theoretical fit using a powder-pattern hneshape.

The time-dependent theoretical point of view provides a simple interpretation of the repetitive pattern. In the time domain, the overlap as a function of time for a given mode oscillates. The separation between the recurrences is a vibrational period. The total overlap is the product of the overlaps of each... 

The damping factor, F, profoundly affects the spectrum. If F is much larger than the highest frequency, recurrences in the overlap will be totally damped out and the spectrum will consist of a broad unstructured band. If F is larger than the difference in the frequencies, the modulation will be damped out but the first recurrence is still observed. The corresponding spectrum will show vibronic structure with the MIME frequency [91-93]. If F is less than the difference between the two frequencies, successive recurrences are not damped out and a modulation will appear in the overlap. The Fourier transform of the overlap in the time domain, giving the spectrum in the frequency domain, is a repeating pattern of bands. The separation between the bands is the difference in frequencies. 

The change of the position of the particles affects the phases and thus the fine structure of the diffraction pattern. So the intensity in a certain point of the diffraction pattern fluctuates with time. The fluctuations can be analyzed in the time domain by a correlation function analysis or in the frequency domain by frequency analysis. Both methods are linked by Fourier transformation. 

Figure 9. Simulated homonuclear DQ MAS spinning-sideband patterns generated in the time domain using eq 6, with the powder average being performed numerically, for different values of the product of D and Trcpi.

Figure 31 presents experimental H DQ MAS spinning sideband patterns for the aromatic protons in (a) the crystalline and (b) the LC phases of a-deuterated HBC—C12.22 The MAS frequency was 35 and 10 kHz in (a) and (b), respectively, with two rotor periods being used for excitation/reconversion in both cases, such that rrcpi equals 57 and 200 /us in the two cases. The dotted lines represent best fit spectra simulated using the analytical time-domain expression for an isolated spin pair in eq 6. As noted in section VIIB, the aromatic protons exist as well isolated pairs of bay protons, and, thus, an analysis based on the spin-pair approximation is appropriate here. As is evident from the insets on the right of Figure 31, the DQ MAS spinning sideband patterns are very sensitive to the product of the D and rrcpi. The best-fit spectra for the solid and LC phases then correspond to DI(Zji)s equal to 15.0 0.9 and 6.0 0.5 kHz, respectively. 

In the following section the power of the fractional derivative technique is demonstrated using as example the derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of the microstructure of disordered media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. 

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