Full aliasing

Aliasing is a term for choosing slower digitization than required by the Nyquist-theorem to cover the full spectral window, while sign discrimination is accomplished by simultaneous acquisition of real and imaginary phase components (called also complex acquisition) [10, 12]. Sequential acquisition for sign discrimination (either TPPI, or the Redfield-Kunz... 

Fig. 1. Schematic 2Q (a) and 3Q (b) correlation maps for an ambient AMPX spin system. Direct correlation peaks are denoted with filled circles, while remote peaks are shown with open circles. Full remote dimensions are presented with no aliasing or folding.

It might be supposed that a smaller design that permitted the estimation of the coefficients of interest could be constructed by taking a fraction of the full factorial. However, the aliasing of three-level designs is very complex and so fractionating a three-level design will not be pursued. The interested reader may refer to Kempthome [18]. 

The interaction between the kinetics, the initial composition and the measurements would be taken into account in a full analysis. This is important for a complete aliasing of kinetics can arise under special circumstances. For example, in a continuum of reactions [22] all of which were first order and irreversible, we might denote the fraction of material with rate constant in the interval (/c, k + dk) at time t by x(t, k) Xo(k) = Jt(0, k). Then for parallel first order reaction we would have... 

Because a fractional factorial design uses less than the complete set of factorial runs, not all of the parameters in the complete model supported by the full factorial can be uniquely estimated. Effect estimates are linked together through aliases. For example, in the 24-1 design in Table 1, the aliases are... 

The full set of de-aliasing basis functions is denoted and contains M functions. The weights W g are assigned based on a least squares fitting procedure. A similar scheme may be constructed for the exchange operator K. By careful control of the grid size, the... 

The full factorial design may not be practical. Perhaps it is not possible for one operator to run two assays on one day. Then a fractional factorial design can be used. The design shown in Table 5 is resolution III design with four experimental trials. As machine is aliased with the interaction between operator and day, the main effect due to machine can be estimated only if we assume there is negligible interaction between operator and day. Often, interactions can be assumed to be negligible based on prior knowledge or the science that is known about the situation. 

Step 3 Choose the design for your experiment. Fractional factorial designs or low-resolution designs are best for process development work where there are several (say four or more) factors to consider. Full factorial designs are used when it is necessary to eliminate all confounding or aliases between main effects and interactions. 

Step 8 If necessary, add more runs to the design matrix to eliminate aliases or confounding patterns. For example, if a fractional factorial experiment shows evidence of interactions between variables, it may be necessary to run the full factorial to determine which interactions are truly important. 

As the DOE is a full-factorial design, there are no aliases to worry about. 

Full details of the origin and effects of leakage and aliasing may be found in several texts.35-37... 

This particular block of eight runs was generated by aliasing D with AB and also E with BC, after carrying out full 2 experiments of A, B, and C. As can be seen from Table 2.5, the best yield from this principal block of experiments, which contains variables and variable interactions expected to be important, was 78 per cent, a considerable improvement over the previously found best yield of 57 per cent. Having identified important factors, or combinations of factors with which they are aliased, it is possible to choose other treatment combinations which will clarify the situation. The best yield obtained for this synthesis was 90 per cent using treatment combination e. 

The widespread use of pointer and reference types makes it impossible in practice to perform full static analysis except on small snippets of code because of the potential for aliasing. It is not possible to avoid reference types in Java, and it is only possible to avoid pointers and references in C++ if polymorphism is not used. In order to perform useful analysis of larger sections of code, it is necessary to make sweeping assumptions to limit the extent of aliasing. While these assumptions may frequently hold, this approach cannot be justified in safety-critical work. 

One of the most important concepts in fractional factorial design is confounding or aliasing. Confounding occurs when two or more interactions share the same column space, that is, the column entries for the interactions are the same. Only a full factorial experiment does not have confounding. By reducing the number of experiments performed, not all of the parameters can be estimated. In a p- ractional experiment. 

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