Convolution relations

For a linear system like a probe circuit, the response fq(f) (RF field created by the coil) to the excitation v(t) (RF voltage produced by the spectrometer) is expressed by a convolution relation... 

We see that transmittance obeys the convolution relation for instrumental... 

We see that absorptance obeys the convolution relation for instrumental spreading under the same conditions of slowly varying BT(x) and DT(x). 

We note here the similarity between the terms absorptance and absorbance. The latter is frequently employed in chemical infrared spectroscopy to denote logi0[l/T(x)], and may also be called optical density. As discussed in Section I.E.2, absorbance may not be directly employed in the convolution relation for instrumental spreading. 

Appendix Some Useful Fourier Transforms and Convolution Relations A.l. Fourier Transforms A.2. Convolution Relations... 

APPENDIX SOME USEFUL FOURIER TRANSFORMS AND CONVOLUTION RELATIONS... 

Mahon PJ, Oldham KB (2006) An examination of the conditions resulting in a temporal convolution related to electrochemical processes. 1 Solid State Electrochem 10(10) 785-791. doi 10.1007/s 10008-006-0169-0... 

Indeed, while the overall dielectric strength, Aetot> is given by the sum of the contributions coming from the two phases, this simple linear convolution relation does not apply for the relaxation time, which instead depends also on the shape, position and relative intensity of the peaks related to relaxation processes of the two phases [18, 19]. For this reason we investigated the kinetics of MROL to OL transformation by analyzing Aen as displayed in Fig. 10.7. 

The last relation in equation (Al.6.107) follows from the Fourier convolution theorem and tlie property of the Fourier transfonn of a derivative we have also assumed that E(a) = (-w). The absorption spectmm is defined as the total energy absorbed at frequency to, nonnalized by the energy of the incident field at that frequency. Identifying the integrand on the right-hand side of equation (Al.6.107) with the total energy absorbed at frequency oi, we have... 

Charge transfer reactions at ITIES include both ET reactions and ion transfer (IT) reactions. One question that may be addressed by nonlinear optics is the problem of the surface excess concentration during the IT reaction. Preliminary experiments have been reported for the IT reaction of sodium assisted by the crown ether ligand 4-nitro-benzo-15-crown-5 [104]. In the absence of sodium, the adsorption from the organic phase and the reorientation of the neutral crown ether at the interface has been observed. In the presence of the sodium ion, the problem is complicated by the complex formation between the crown ether and sodium. The SH response observed as a function of the applied potential clearly exhibited features related to the different steps in the mechanisms of the assisted ion transfer reaction although a clear relationship is difficult to establish as the ion transfer itself may be convoluted with monolayer rearrangements like reorientation. 

This is the definition of a convolution (p. 16, Eq. (2.17)) of the distribution hi x) with itself. Repeated induction yields the relation... 

We start by creating a matrix. This matrix is based on the index of coefficients that are to be ultimately produced. Savitzky and Golay labeled the coefficients in relation to the central data point of the convolution, therefore a three-term set of coefficients are labeled -1, 0, 1. A five-term set is labeled -2, -1, 0, 1, 2 and so forth. 

Our new method of determining nonlinearity (or showing linearity) is also related to our discussion of derivatives, particularly when using the Savitzky-Golay method of convolution functions, as we discussed recently [6], This last is not very surprising, once you consider that the Savitzky-Golay convolution functions are also (ultimately) derived from considerations of numerical analysis. 

The degree of activation of the sample is measured by post-irradiation spectroscopy, usually performed with high-purity semiconductors. The time-resolved intensity measurements of one of the several spectral lines enables to get the half-life of the radioactive element and the total number of nuclear reactions occurred. In fact, the intensity of a given spectral line associated with the decay of the radioactive elements decreases with time as Aft) = Aoexp[—t/r], where Aq indicates the initial number of nuclei (at t = 0) and r is the decay time constant related to the element half-life (r = In2/ /2), which can be measured. Integrating this relation from t = 0 to the total acquisition time, and weighting it with the detector efficiency and natural abundance lines, the total number of reactions N can be derived. Then, if one compares this number with the value obtained from the convolution of... 

The convolution integral and the Exponential Piston Flow Model (EPM) were used to relate measured tracer concentrations to historical tracer input. The tritium input function is based on tritium concentrations measured monthly since the 1960s near Wellington, New Zealand. CFC and SF6 input functions are based on measured and reconstructed data from southern hemisphere sites. The EPM was applied consistently in this study because statistical justification for selection of some other response function requires a substantial record of time-series tracer data which is not yet available for the majority of NGMP sites, and for those NGMP sites with the required time-series data, the EPM and other response functions yield similar results for groundwater age. 

One of the main limitations of the experimental methodology described above is related to the time constraint. It hinders the study of many interesting reactions that are too slow to ensure that the amplitude of the photoacoustic wave is independent of the kinetics of the process. This is the case, for instance, of transient lifetimes in the range of 100 ns to 0.1 ms for a 0.5 MHz transducer. Fortunately, there is an alternative procedure to deal with those cases where the condition r 1/v does not hold. The procedure, known as time-resolved PAC (TR-PAC), was developed by Peters and co-workers [282,284,299] and considers that the observed wave, Sexp(t), reflects the kinetics of the true heat deposition, S(t), as well as the detector response wave, T(t). In other words, SexP(t) is the convolution of S(t) with the transducer function, T(t)... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

A more recent example of this technique has been the study on human absorption characteristics of fexofenadine [109], Fexofenadine has been shown to be a substrate for P-gp in the in vitro cell lines its disposition is altered in knockout mice lacking the gene for MDRla, and co-administration of P-gp inhibitors (e.g. ketoconazole and verapamil) was shown to increase the oral bioavailability of fexofenadine [110-113], Hence, it is suggested that the pharmacokinetics of fexofenadine appears to be determined by P-gp activity. In the human model, the intestinal permeability estimated on the basis of disappearance kinetics from the jejunal segment is low, and the fraction absorbed is estimated to be 2% [114], Co-administration of verapamil/ketoconazole did not affect the intestinal permeability estimates however, an increased extent of absorption (determined by de-convolution) was demonstrated. The increased absorption of fexofenadine was not directly related to inhibition of P-gp-mediated efflux at the apical membrane of intestinal cells as intestinal Peff was unchanged. Furthermore, the effect cannot be explained by inhibition of intestinal based metabolism, as fexofenadine is not metabolised to any major extent. It was suggested that this may reflect modulation of efflux transporters in hepatocyte cells, thereby reducing hepatobiliary extraction of fexofenadine. 

---