Zero filling method

All the processing steps in this section are intended to enhance the time domain data leading to the suppression of distortions or artefacts and an improvement in the overall spectral quality. The suitability of the methods can be estimated from the appearance of the FID in the main spectrum window of ID WIN-NMR as illustrated in the schematic FID shown in Fig. 3.6. The envelope of the exponentially decaying FID has a dc offset as it is not symmetrical about the zero line whilst the spectrum lineshape may be improved by either zero filling or apodization or possibly both. 

While methods of spectrum analysis capable of super-resolution exist, that is, methods that can achieve resolution greater than l/Wx, the most common of these, linear prediction (LP) extrapolation, has substantial drawbacks. LP extrapolation is used to extrapolate signals beyond the measured interval. While this can dramatically suppress truncation artifacts associated with zero-filling as well as improve resolution, because LP extrapolation implicitly assumes exponential decay it can lead to subtle frequency bias when the signal decay is not perfectly exponential [8]. This bias can have detrimental consequences for applications that require the determination of small fi-equency differences, such as measurement of residual dipolar couplings (RDCs). 

The method of zero filling. In this method, spectral interpolation is derived as a result of processing a measured interferogram. As is clear from Equation (5.3), the number of data points in the spectrum obtained by the discrete Fourier transform is equal to the number of data points in the interferogram before the transform. If zero points are added... 

Homonuclear 2D correlation (COSY) spectra were acquired according to the method of Nagayama et al, and Homonuclear Hartmann-Hahn experiment (HOHAHA) according to Bax. The 2D data were zero filled and weighted prior to Fourier transformation as appropriate. A sine-bell filtering function was used in both dimensions. 

Solving several equations by the method of Gaussian elimination, one might divide the first equation by [ j. obtaining 1 in the [ j position. Multiplying ayi into the first equation makes an ay]. Now subtracting the first equation from the second, a zero is produced in the a2i position. The same thing can be done to produce a zero in the a. ] position and so on, until the first column of the eoefflcient matrix is filled with zeros except for the Un position. 

However, even with the most advanced measuring and simulation tools, the most efficient methods are simple calculations that give an order-of-magnitude estimation of the influence of a phenomenon. Time constants for diffusion, heat conduction, and acceleration are very useful. For example, the time constant for diffusion Td = f/D is the time it takes to fill a cube of size I by diffusion, and the time for a particle to accelerate from zero velocity to approximately two-third of the velocity of the surrounding fluids is 118/j, where p[Pg.331]

An alternative to the measurement of the dimensions of the indentation by means of a microscope is the direct reading method, of which the Rockwell method is an example. The Rockwell hardness is based on indentation into the sample under the action of two consecutively applied loads - a minor load (initial) and a standardised major load (final). In order to eliminate zero error and possible surface effects due to roughness or scale, the initial or minor load is first applied and produce an initial indentation. The Rockwell hardness is based on the increment in the indentation depth produced by the major load over that produced by the minor load. Rockwell hardness scales are divided into a number of groups, each one of these corresponding to a specified penetrator and a specified value of the major load. The different combinations are designated by different subscripts used to express the Rockwell hardness number. Thus, when the test is performed with 150 kg load and a diamond cone indentor, the resulting hardness number is called the Rockwell C (Rc) hardness. If the applied load is 100 kg and the indentor used is a 1.58 mm diameter hardened steel ball, a Rockwell B (RB) hardness number is obtained. The facts that the dial has several scales and that different indentation tools can be filled, enable Rockwell machine to be used equally well for hard and soft materials and for small and thin specimens. Rockwell hardness number is dimensionless. The test is easy to carry out and rapidly accomplished. As a result it is used widely in industrial applications, particularly in quality situations. 

The permeation technique is another commonly employed method for determining the mutual diffusion coefficient of a polymer-penetrant system. This technique involves a diffusion apparatus with the polymer membrane placed between two chambers. At time zero, the reservoir chamber is filled with the penetrant at a constant activity while the receptor chamber is maintained at zero activity. Therefore, the upstream surface of the polymer membrane is maintained at a concentration of c f. It is noted that c f is the concentration within the polymer surface layer, and this concentration can be related to the bulk concentration or vapor pressure through a partition coefficient or solubility constant. The amount... 

Another method to measure pore size distribution is capillary flow porometry [202,203], in which a sample material is soaked with a low surface tension liquid that fills all its pores. Then, gas pressure is applied on one side of the sample in order to force the liquid out of the pores. At low pressures, the flow rate is close to zero however, as the pressure increases, the flow rate also increases and the amount of liquid inside fhe pores decreases. Thus, the flow rate is determined as a function of pressure and is then used to calculate the desired pore characteristics, such as pore size distribution, largest pore diameter, and mean flow pore diameter. 

---