8 step function resolution

Figure 23 Calculation of the shape of the actively compensated pulse can be carried out on the software. (A) shows the real (red line) and the imaginary (green line) component of an example of the target pulse shape t>,(f). Its leading and the trailing edges have a cosine shape with a transition time of 1.25 xs in 50 steps, and the width of the plateau is 5 ps. (B) Laplace transformation B(s) multiplied by the Laplace transformed step function U(s). (C) It was then divided by the Laplace transformation Y(s) of the measured step response y(t) of the proton channel of a 3.2-mm Varian T3 probe tuned at 400.244 MHz to obtain V(s). (D) Finally, inverse Laplace transformation was performed on V(s) to obtain the compensated pulse that results in the RF pulse with the target shape. Time resolution was 25 ns, and o = 20 was used for the Laplace and inverse Laplace transformations.

Next, bi(t) was Laplace transformed into B(s), and then multiplied by the Laplace transformation U(s) of the step function u(t). The result B(s)U(s) is displayed in Figure 23B. In this example, the step response y(t) was measured for the 1H channel of a Varian 3.2 mm T3 probe tuned at 400.244 MHz with a time resolution of 25 ns, and Laplace transformed into Y(s). By dividing B(s)U(s) by Y(s), the function plotted in Figure 23C was obtained, from which, by performing inverse Laplace transformation, the programming pulse shape v(t) was finally obtained, as shown in Figure 23D. The amplitude and the phase of the complex function v(t) give the intensity and the phase of the transient-compensated shaped pulse. 

Sometimes, the profile is so short that it cannot be resolved by the measurement technique. Such information may also be applied to constrain cooling rate. For example, if the spatial resolution of the measurement is I, the absence of a profile (i.e., a step-function profile) means that jDdtasymptotic cooling history, then Dqt < f, leading to... 

Example5.18. Watson and Cherniak (1997) reported diffusivity in zircon under wet conditions as D = exp(—25.93 — 25,280/T) m /s. In a natural zircon crystal, oxygen isotope ratio shows a step function across the core and mantle. Suppose the spatial resolution is 5 pm. The initial temperature is constrained independently (e.g., from the mineral assemblage) to be HOOK and the cooling may be assumed to be as5miptotic. Constrain the cooling timescale. 

If ksT is smaller than the energy resolution required in the measurement, then the Fermi distribution function can be approximated by a step function. In this case, the tunneling current is (see Fig. 1.20) ... 

Rather than using short-duration pulsed-light sources, which usually yield information concerning nonequilibrium->equilibrium relaxations, it is often of great interest to study equilibrium- equilibrium relaxations and excitations. This requires the use of step-function light sources. The temporal resolution in this case is determined by the rise time of the step. There are, of course, two types of steps that one need consider (a) a source that is initially off and then is rapidly turned on and maintained on for some time, and (b) a source that is initially on and then is rapidly turned off and then maintained off for some time. 

Fig. 8. Outline of the time-scale of the processes observed during an electron transfer reaction observed through thermal lensing. Processes which occur in times below ca. 0.5 ps are very fast , beyond the temporal resolution of the thermal lensing technique they would appear as a step function in the kinetics of heat release. The slowest processes which would be observed in this case are the second-order recombinations of free ions, which take place in time scales of ps to several ms.

The resolution p Rg(N) already allows us to obtain an explicit measure of the brush layer thickness L (see Fig. 36). In this case the simplest step-function profile ( )(z) with constant composition in the brush layer region is assumed (see Fig. 33). While the de Gennes-Leibler model assumes all end-attached chains to stretch at the same distance z=L from the interface, the situation with a lower free energy is conceivable [226,228] characterized by the non-uniform stretching and the total brush concentration decreasing with z. Measurements performed with higher resolution reveal [242,243,261,264] the profiles ( )(z) of the stretched brushes which might be approximated by an error function [266] ... 

Presentation of the analytical signal in the form of a peak, rather than in the form of rounded-off step function (wave), with consequent increased resolution possibilities. 

Depth resolution is most commonly defined as the depth over which a signal from some abruptly appearing layer climbs from 16% of its maximum intensity to 84% when plotted on a hnear scale as this represents two standard deviations ( lGaussian function with a step function. This is illustrated in Figure 5.5. This definition is also applied to decaying signals from abruptly terminated layers. Caution must, however, be exercised when matrix effects and/or radiation-enhanced segregation are active, as these can modify the value derived relative to the absolute sputter-induced depth resolution. 

Due to limited number of experiments, F (SoQ) and F, y, (SoQ) were assumed as stepped functions, but in reality, they are probably curved functions. More accurate functions can be found by performing aging experiments with a higher resolution of SoC, i.e., conducting long-term storage experiment at SoC = 10, SoC = 20, SoC = 30 and so on. 

In addition to determining the presence of impurities in solid solution, it is also important to study the distribution of such impurities. The importance of knowing how the impurities are distributed, whether completely statistically at the atomic level or in a single inclusion, is vital to every interpretation of properties measured on the phase. In this field, the last ten years have brought a step-function advance with the introduction of the electron microprobe. Yet, even here spatial resolution is limited to about 1 pm, and sensitivity and precision are sometimes insufficient. 

So, a comparison of different types of magnetic field sensors is possible by using the impulse response function. High amplitude and small width of this bell-formed function represent a high local resolution and a high signal-to-noise-characteristic of a sensor system. On the other hand the impulse response can be used for calculation of an unknown output. In a next step it will be shown a solution of an inverse eddy-current testing problem. 

A multidimensional system using capillary SEC-GC-MS was used for the rapid identification of various polymer additives, including antioxidants, plasticizers, lubricants, flame retardants, waxes and UV stabilizers (12). This technique could be used for additives having broad functionalities and wide volatility ranges. The determination of the additives in polymers was carried out without performing any extensive manual sample pretreatment. In the first step, microcolumn SEC excludes the polymer matrix from the smaller-molecular-size additives. There is a minimal introduction of the polymer into the capillary GC column. Optimization of the pore sizes of the SEC packings was used to enhance the resolution between the polymer and its additives, and smaller pore sizes could be used to exclude more of the polymer... 

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