Time course interpolation

According to the idea of Savova-Stoynov and Stoynov (Savova-Stoynov and Stoynov [1985, 1992], Stoynov [1990]), recording of a series of impedance measurements at distinct time intervals offers the possibility to eliminate the drift and therefore to reconstruct an impedance spectrum which is acquired in an infinite short time. The time course interpolation procedure is schematically depicted in... 

Figure 4.5.43. Schematic representation of time-course interpolation.

No serious attempt has yet been made to standardize filtration tests and to categorize filtration behavior in generally accepted terms. A possibly useful measure of filterability, however, has been proposed by Purchas (1977 1981). The time in minutes required to form a cake 1 cm thick when the cell is operated with a differential of 500Torr (0.67 bar) is called the Standard Cake Formation Time (SCFT), tF. The pressure of 500Torr is selected because it is obtained easily with common laboratory equipment. The procedure suggested is to make a series of tests at several cake thicknesses and to obtain the SCFT by interpolation, rather than to interrupt a single test to make observations of cake thickness. A direct relation exists, of course, between the SCFT and resistivity a some examples are... 

The transition from the atom to the cluster to the bulk metal can best be understood in the alkali metals. For example, the ionization potential (IP) (and also the electron affinity (EA)) of sodium clusters Na must approach the metallic sodium work function in the limit N - . We previously displayed this (1) by showing these values from the beautiful experiments by Schumacher et al. (36, 37) (also described in this volume 38)) plotted versus N". The electron affinity values also shown are from (39), (40) and (34) for N = 1,2 and 3, respectively. A better plot still is versus the radius R of the N-mer, equivalent to a plot versus as shown in Figure 1. The slopes of the lines labelled "metal sphere" are slightly uncertain those shown are 4/3 times the slope of Wood ( j ) and assume a simple cubic lattice relation of R and N. It is clear that reasonably accurate interpolation between the bulk work function and the IP and EA values for small clusters is now possible. There are, of course, important quantum and statistical effects for small N, e.g. the trimer has an anomalously low IP and high EA, which can be readily understood in terms of molecular orbital theory (, ). The positive trimer ions may in fact be "ionization sinks" in alkali vapor discharges a possible explanation for the "violet bands" seen in sodium vapor (20) is the radiative recombination of Na. Csj may be the hypothetical negative ion corresponding to EA == 1.2 eV... 

This new differential equation for CP (2) is solved by the SIMULATOR module for the parameters C (2), C (3), and C (4), at default or specified points in time. During the course of this simulation phase, required values for C (1) and CP(1) are calculated using numerical interpolation and differentiation. The CURVEFIT module then solves the equation for CP(1) for K and K2. This curve fitting part also involves a test for mathematical validity, using as its basis the user s estimate of the reliability of the data. 

This property states that stretching a time domain signal causes its spectrum to become narrower, and shortening a time domain signal causes its bandwidth to spread. Of course, care must be taken to properly bandlimit and interpolate the resampled signal. 

As discussed in Section 8.3.8, the confidence limits (CLs) obtained from nonlinear least-squares regression are not very reliable. If time, availability of standards and finances permit, a better feel for the true CLs is obtained by repeating the same experiment (calibration) several times. Similar comments apply to CLs of interpolated values of (Qa /Qsis ) from measured values of (Ra /Rsis) in actual analysis using the nonlinear regression parameters evaluated from the calibration experiments (compare Equation [8.32] for the linear regression case). Of course, as before, the calibrators can in principle be made up either as mixed solutions of pure analytical and internal standards or as extracts of spiked matrix blanks. 

These relations offer the possibility to examine measured transfer functions (impedance spectra) on errors caused by time instability or time drift. However, KK-checking techniques have fundamental problems in their application to practical measurements. Therefore many attempts have been made to overcome these limitations by means of different interpolation procedures. An attempt is the Z-HIT approximation, applied by Schiller et al. [2001], Agarwal et al. [1995], and Ehm et al. [2000], an approximation formula for the calculation of the impedance modulus course from the phase angle by integration. 

No major problems were encountered in performing PLS at the various time scales. Linear interpolation was used for the chip and pulp quality data at the I-hour time scale. The fit of all the models was reasonably good, with values between 0.66 and 0.92, and values of 0.56 to 0.72. Of course, all the runs were performed on only one week of data, meaning that the models ability to predict other weeks is probably much lower. 

More elaborate formulas have been derived by Schwarzl for calculating J(t) from one value of 7 at w = 1 ft and several values of J" at frequencies equally spaced logarithmically from co/16 to 8co, or alternatively from one value of J" at CO = 1 // and several values of J at frequencies equally spaced logarithmically from co/4 to 64co. The error bounds of the approximation have been carefully examined. This type of calculation is easily performed by computer if smoothed interpolated data for 7 and 7" are available. In either the Ninomiya or the Schwarzl method, 7(0 is of course obtained over a somewhat narrower time range than the frequency range of the original data. 

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