Time-frequency Fourier transform

After processing in the time domain, Fourier transformation, phasing and basic processing (calibration, peak picking, integration) ahs been performed, additional processing steps to improve spectral quality are at your disposal. This includes operations common to both ID and 2D spectra e.g. baseline correction in the frequency domain, as well as operations specific to these different type.s of data sets. 

The data flux of a two-dimensional carbon-proton shift correlation is similiar to that described in Fig. 2.50(a) for a. /-resolved 2D CMR experiment, with one difference Instead of carbon-proton couplings JCH, proton chemical shifts (iH are stored in the evolution time tl. Fourier transformation in the (2 domain thus yields a series of NMR spectra with carbon-13 signals modulated by the attached proton Larmor frequencies. A second Fourier transformation in the domain generates the dH, Sc matrix of a two-dimensional carbon-proton correlation. 

In contrast to the diffusion processes, the fluctuating force 3F t) can no longer be derived from the Wiener process, and its spectral density defined as 2kBT 0 times the Fourier transform of the memory function C(f) is frequency limited and has no more the white noise characteristics. 

The microdielectrometer has been used to monitor polymer curing in-situ 22.23. The CFT device can make measurements at lower frequencies than could be achieved by conventional dielectric measurement techniques. Measurements at multiple frequencies can be made in real-time. A Fourier transform equivalent of the microdielectrometer has been developed to extend the frequency range to as low as 0.005 Hz 24. 

Equation (9.5) can be viewed as first a modulation of the window to frequency cq thus producing a bandpass filter w(n)eJ an followed by a filtering of x(n) through this bandpass filter. The output is then demodulated back down to baseband. The temporal output of the filter bank can be interpreted as discrete sine waves that are both amplitude- and phase-modulated by the time-dependent Fourier transform. 

We shall conclude this chapter with a few speculative remarks on possible future developments of nonlinear IR spectroscopy on peptides and proteins. Up to now, we have demonstrated a detailed relationship between the known structure of a few model peptides and the excitonic system of coupled amide I vibrations and have proven the correctness of the excitonic coupling model (at least in principle). We have demonstrated two realizations of 2D-IR spectroscopy a frequency domain (incoherent) technique (Section IV.C) and a form of semi-impulsive method (Section IV.E), which from the experimental viewpoint is extremely simple. Other 2D methods, proposed recently by Mukamel and coworkers (47), would not pose any additional experimental difficulty. In the case of NMR, time domain Fourier transform (FT) methods have proven to be more sensitive by far as a result of the multiplex advantage, which compensates for the small population differences of spin transitions at room temperature. It was recently demonstrated that FT methods are just as advantageous in the infrared regime, although one has to measure electric fields rather than intensities, which cannot be done directly by an electric field detector but requires heterodyned echoes or spectral interferometry (146). Future work will have to explore which experimental technique is most powerful and reliable. 

Finally, again using the facts that Gci(t) is even in time and that one can subtract any constant from Gci(t) within the (finite-frequency) Fourier transform without affecting the result, we obtain... 

An interferogram is a spectrum in the time domain. Fourier transformation converts it to the frequency domain. 

Ti) and spin-spin (T2) relaxation time constants. Fourier transformation of the time domain data produces a frequency spectrum where the amplitude at each frequency is a measure of the number of nuclei in the corresponding region in space. In magnetic resonance imaging a multivariate slice for one pulse sequence is used if spatial resolution is most important. A full NMR spectrum can be produced for a single point inside the material if the spectral aspect is most important. 

Note that there is no generally agreed convention on the sign in exponentials in the forward and inverse Fourier transform, other than the forward and inverse transformations have opposite signs. By analogy with time-domain signals, the (m, w) coordinates in the Fourier domain are called spatial frequencies. Fourier transforms of real functions will generally be complex-valued. A discrete version of the Fourier transform (for sampled data) also exists. 

The method proposed by Papoulis [7] to determine h(t) as a function of its Fourier transform within a band, is a non-linear adaptive modification of a extrapolation method.[8] It takes advantage of the finite width of impulse responses in both time and frequency. 

While the data are collected in the time domain by scaiming a delay line, they are most easily interpreted in the frequency domain. It is straightforward to coimect the time and frequency domains tln-ough a Fourier transform... 

In practical applications, x(t) is not a continuous function, and the data to be transformed are usually discrete values obtained by sampling at intervals. Under such circumstances, I hi discrete Fourier transform (DFT) is used to obtain the frequency function. Let us. suppose that the time-dependent data values are obtained by sampling at regular intervals separated by [Pg.43]

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

In order to analyze the vibrations of a single molecule, many molecular dynamics steps must be performed. The data are then Fourier-transformed into the frequency domain to yield a vibrational spectrum. A given peak can be selected and transformed back to the time domain. This results in computing the vibra-... 

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