Time-domain sensing

As mentioned earlier, we acquire data in the time domain but to make sense of it, we need to view it in the frequency domain. This is where the Fourier transformation comes in. There is not too much to do here - there are no parameters to change, ft is a necessary step but the automatic routines will perform this for you with no input. 

FRET to fluorescent acceptor is obviously more popular because of its two-channel self-calibrating nature. Sensing may result in switching between two fluorescent states, so that in one of them a predominant emission of the donor can be observed and in the other - of the acceptor. This type of FRET can be extended to time domain with the benefit of using simple instrumentation with the longlifetime donors [27]. 

B) up-down HMBC pulse sequence inverting BCH and 13CH3 peaks relative to the standard sequence. (C) up-down + HMBC pulse sequence inverting 13CH and 13CH3 peaks relative to the standard sequence and in the opposite sense to the up-down sequence. The data of the different pulse sequences are recorded in an interleaved manner. After formation of the required two linear combinations in the time domain, the data are processed in the same way as other HMBC-type data. 

Figure 1.1. Schemes for fluorescence sensing intensity, intensity ratio, time-domain, and phase-modulation, from left to right.

Figure 1.2. Intensity, time-domain, and frequency-domain sensing, as applied in the laboratory, a cuvette, and blood sample in a clinical setting,...

Figure 13.5. Methods of fluorescence sensing, (a) Single excitation/emission wavelength intensity changes with analyte concentration (b) wavelength-ratiometric A/B changes with analyte concentration (c) liftetime based (time domain) r changes with analyte concentration (d) lifetime-based (phase modulation) Am and A change with analyte concentration.

Studying the dynamics of systems in the time domain involves direct solutions of differential equations. The computer simulation techniques of Part II are very general in the sense that they can give solutions to very complex nonlinear problems. However, they are also very specific in the sense that they provide a solution to only the particular numerical case fed into the computer. 

The term 2D NMR, which stands for two-dimensional NMR, is something of a misnomer. All the NMR spectra we have discussed so far in this book are two dimensional in the sense that they are plots of signal intensity versus frequency (or its Fourier equivalent, signal intensity versus time). By contrast, 2D NMR refers to spectroscopic data that are collected as a function of two time scales, tx (evolution and mixing) and t2 (detection). The resulting data set is then subjected to separate Fourier transformations of each time domain to give a frequency-domain 2D NMR spectrum of signal intensity versus two frequencies, Fx (the Fourier transform of the t time domain) and F2 (the Fourier transform of the t2 time domain). Thus, a 2D NMR spectrum is actually a three-dimensional data set ... 

The above integral exists in the sense of having a Cauchy principal value for every value of w. Conversely if we have the Fourier transform, then we can construct the function in the time-domain from,... 

In a general sense, the frequency-domain error structure is determined by the nature of errors in the time-domain signals and by the method used to process the time-domain data into the frequency domain. The ceU impedance influences the frequency-dependence of the variance of the measurements, but the cell impedance does not influence whether the variances of real and imaginary components are equal or whether errors in the real and imaginary components are uncorrelated. 

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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