Exponential decay function Fourier transform

Fig. 7 PELDOR signal analysis, (a) Time domain PELDOR signal as a function of the delay time T of the pump pulse. The dashed line shows the exponentially decaying intermolecular dipolar contribution to the signal, (b) Time domain PELDOR signal after division of the original PELDOR time domain data by the fit-function representing the intermolecular decay, (c) Fourier transform of the PELDOR time trace (b) representing the dipolar Pake-pattem. (d) Distance distribution function obtained from the PELDOR time traces (b) by Tikhonov regularization. From the last representation the distances for spin pairs A-B can be the most easily extracted...

Where eo and Soo are the limiting low and high frequency real permittivities, and indicates a one-sided Fourier transformation. A simple form for (.t) is the single exponential decay function... 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

Platinum and palladium porphyrins in silicon rubber resins are typical oxygen sensors and carriers, respectively. An analysis of the characteristics of these types of polymer films to sense oxygen is given in Ref. 34. For the sake of simplicity the luminescence decay of most phosphorescence sensors may be fitted to a double exponential function. The first component gives the excited state lifetime of the sensor phosphorescence while the second component, with a zero lifetime, yields the excitation backscatter seen by the detector. The excitation backscatter is usually about three orders of magnitude more intense in small optical fibers (100 than the sensor luminescence. The use of interference filters reduce the excitation substantially but does not eliminate it. The sine and cosine Fourier transforms of/(f) yield the following results ... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

Fig. 2. (a,b) Transient absorption on the v0h=1— 2 transition of OH/OH dimers (symbols). The spectrally integrated, anisotropy free absorption change AA is plotted as a function of the delay time between the pump centered at Ep=2950 cm 1 and the probe centered at Epr. Solid lines exponential decay with a time constant of 200 fs. Inset of Fig. (b) Fourier transform of the oscillatory component of the transient in Fig. (b) displaying an oscillation frequency of 145 cm 1. 

Fig. 2. (a) The free induction decay, G(t) for 19F in a single crystal of CaFi for B0 along [1,0,0]. The experimental points are given by circles and crosses from the CW and pulse measurements, respectively, and the theoretical curve is that of Eq. (14), corresponding to an exponential decay multiplied by a sine function. Note that F(t) is equivalent to G(t) in the present notation. Reproduced with permission from A. Abragam, The Principles of Nuclear Magnetism, p. 121, Oxford University Press, London, 1961. (b) The lineshape in the frequency domain corresponding to the Fourier transform of the theoretical curve. 

In the oxygen VER experiments (3) the n = 1 vibrational state of a given oxygen molecule is prepared with a laser, and the population of that state, probed at some later time, decays exponentially. Since in this case tiojo kT, we are in the limit where the state space can be truncated to two levels, and 1/Ti k, 0. Thus the rate constant ki o is measured directly in these experiments. Our starting point for the theoretical discussion is then Equation (14). For reasons discussed in some detail elsewhere (6), for this problem we use the Egelstaff scheme in Equation (19) to relate the Fourier transform of the quantum force-force time-correlation function to the classical time-correlation function, which we then calculate from a classical molecular dynamics computer simulation. The details of the simulation are reported elsewhere (4) here we simply list the site-site potential parameters used therein e/k = 38.003 K, and a = 3.210 A, and the distance between sites is re = 0.7063 A. 

Equations (2)-(5) represent a complete recipe for calculating the response of the spin system to a m.p. sequence. The spectrum is obtained, as in the real experiment, by a discrete Fourier transformation of the time series after it had been multiplied by a suitable filter function, for example, a decaying exponential. 

Fourier transformation of a time-domain signal f(jt) decaying with time constant T-i in an exponential fashion Such a signal is the NMR impulse-response function which can be derived from the Bloch equations (Fig. 4.1.1(b), cf. eqn. 2.2.19),... 

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