Patterns frequency domain

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. 

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. 

Figure 1.2 From the masking pattern it can be seen that the excitation produced by a sinusoidal tone is smeared out in the frequency domain. The right hand slope of the excitation pattern is seen to vary as a function of masker intensity (steep slope at low and flat slope at high intensities).

Figure 1.4 Excitation pattern for a short tone burst. The excitation produced by a short tone burst is smeared out in the time and frequency domain.

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). 

In order to determine couplings to nuclei in natural abundance, it is necessary to suppress the signals of protons that are not bonded to a magnetically active heteronucleus. An (iix hetero half-filter that selects for such nuclei in F, via the phase cycle was used for this purpose. Presaturation of the protons bonded to 12C by the BIRD pulse allows a rapid pulse-sequence (2 scans per second). The resulting 2D spectra are TOCSY spectra in which the cross peaks show the desired E. COSY pattern. From the results shown, the only limitation seems to be the resolution obtained, although the authors do not hesitate to use a third heteronuclear frequency domain for improvement. 

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.

Division of the observed fluorescence fluctuation spectrum (Fig. 8b) by such a pattern yields the frequency domain lifetime directly through (50). This is shown for Rhodamine B in Fig. 9 to yield a decay time of 3.2 ns. 

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 3 Sampling in the frequency domain (a) Modulated signal, x has frequency spectrum Xp (b) Harmonics of the carrier signal (c) Spectrum of modulated signal is a repetitive pattern of Xp and Xpcan be completely recovered by low pass filtering using, for example, a box filter with cut-offfrequency (d) Too low a...

The peak search by 2nd derivatives represents a kind of sharpening (deconvolution), i.e. a division by the Fourier transform of a certain peak shape in the frequency domain. This is possible only if this Fourier transform has no zeroes, i.e. if it monotonically approaches zero. Bromba and Ziegler (1984) report such an algorithm, but its usability for X-ray patterns was not proved until now. 

Basis states are equally important in understanding frequency- and time-domain experiments. In the frequency domain, the pattern is comprised of the... 

As the X-ray energy is increased beyond that required for promotion of the core electron, ejection of core electrons into the continuum occurs. This ejected electron propagates from the Mn center until it encounters another atom from which it can be back-scattered. The interference of back-scattered waves with propagating waves leads to an interference pattern that is manifested as an oscillation in the X-ray absorption pattern. Fourier transformation of this oscillating spectrum from the frequency domain to the distance domain gives a new spectrum whose abscissa contains information on the distance between the target atom (i.e., the Mn center) and the back-scattering atoms. This second technique is called Extended X-ray Absorption Fine Structure, EXAFS, and has been the only spectroscopic tech-... 

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