Time domain significance

Do not worry if you have forgotten the significance of the characteristic equation. We will come back to this issue again and again. We are just using this example as a prologue. Typically in a class on differential equations, we learn to transform a linear ordinary equation into an algebraic equation in the Laplace-domain, solve for the transformed dependent variable, and finally get back the time-domain solution with an inverse transformation. 

It has been shown recently, however, that these equations may be solved 62), by means of the state functions theory (64) and/or the time-domain matrix methods 63). Figure 14 shows, for instance, that the computer calculations allow us to determine, with a good approximation, the time-dependence of thermal phenomena taking place in the calorimeter, although all significant details of their kinetics are completely blurred on the thermogram 62). This method has been recently used to correct... 

There is significant debate about the relative merits of frequency and time domain. In principle, they are related via the Fourier transformation and have been experimentally verified to be equivalent [9], For some applications, frequency domain instrumentation is easier to implement since ultrashort light pulses are not required, nor is deconvolution of the instrument response function, however, signal to noise ratio has recently been shown to be theoretically higher for time domain. The key advantage of time domain is that multiple decay components can, at least in principle, be extracted with ease from the decay profile by fitting with a multiexponential function, using relatively simple mathematical methods. 

When the internal motion is so rapid that it relaxes before significant relaxation by twisting and bending takes place, there exists a time domain in which 7 (t)s/n(oo)yet C (t) = 1.0, F (t)s 1.0, and the residual anisotropy is given by... 

In principle, Mq can be determined through the first point in time domain for t = 0 or by integration of L. However, under experimental conditions distortions in the acquired time signal (e.g., due to eddy currents) are transferred to the frequency domain by Fourier transformation and can result in significant differences in quantification. A straightforward quantification method in the frequency domain is simply to determine the total integrated area under a resonance in a distinct frequency range of the spectrum. This method works well for spectra with well separated resonance lines and without... 

The main experimental elfects are accounted for with this model. Some approximations have been made a higher-level calculation is needed which takes into account the fact that the charge distribution of the trapped electron may extend outside the cavity into the liquid. A significant unknown is the value of the quasi-free mobility in low mobility liquids. In principle, Hall mobility measurements (see Sec. 6.3) could provide an answer but so far have not. Berlin et al. [144] estimated a value of = 27 cm /Vs for hexane. Recently, terahertz (THz) time-domain spectroscopy has been utilized which is sensitive to the transport of quasi-free electrons [161]. For hexane, this technique gave a value of qf = 470 cm /Vs. Mozumder [162] introduced the modification that motion of the electron in the quasi-free state may be in part ballistic that is, there is very little scattering of the electron while in the quasi-free state. 

Although phase-encoding solves the problems associated with large time-domain dyanmic range in SWIFT excitation, the final frequency-domain spectrum still exhibits noticeable oscillation in magnitude at frequencies near the limits of the bandwidth. The final spectrum can be smoothed significantly by prior weighting ("apodization") of the... 

As has been described, CD is one of the few spectroscopic techniques sensitive to the structural parameters that define and guide the three-dimensional shape of a biologically active entity. This important structural information, obtained on a time-resolved basis in the same time domain as the biological event under study, is an extremely important advance that will have wide application in biophysical studies of proteins, polypeptides and other biologically significant species. 

The application of multichannel light detection devices to the measurement of CD will significantly improve the technique and allow extension into areas not presently accessible by conventional, sequentially scanned CD. For example, multichannel devices will enable SNR improvements via spectral averaging. Alternately, a complete CD spectrum can be obtained on a transient species in the time domain required for a single wavelength... 

The matter of sampling and limited representation of frequencies requires a second look at the representation of data in the time and frequency domains, as well as the transformation between those domains. Specifically, we need to consider the Fourier transformation of bandwidth-limited, finite sequences of data so that the S/N enhancement and signal distortion of physically significant data can be explored. We begin with an evaluation of the effect of sampling, and the sampling theorem, on the range of frequencies at our disposal for some set of time-domain data. 

FIGURE 10.12 Time-domain smoothing of the noisy data in Figure 10.1 with the impulse response function of Figure 10.7, processed from left to right in this spectrum. The true signal is shown as a dotted line. Note the significant filter lag in this example. 

The spectral width of a pulse train emitted by a femtosecond laser can be significantly broadened in a single mode fiber [27]. This process that maintains the mode structure is described in the time domain by the optical Kerr effect or selfphase modulation. The first discussion is simplified by assuming an unchanging pulse-shape under propagation. After propagating the length l the intensity dependent refractive index n(t) = n0 + ri2/(f) leads to a self induced phase shift... 

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