Fourier decomposition technique

Fourier-based techniques. For these methods, the input parameter realisations have to fulfil special frequency properties. Then a fi quency decomposition of the output maps different frequencies attributed to the input factors to different fractions of the variance of the output. Different frequency selection schemes have been developed, named Fourier Amplitude Sensitivity Test (FAST), Extended FAST (EFAST), and Random Balance Design (RBD). They can estimate first and/or total effects. 

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

Since we deal with a periodic pattern, it is possible to apply a technique that was originally invented by the French physicist and mathematician Jean Baptiste Joseph Fourier (1768-1830). Fourier was the first who showed that every periodic process (or an object like in our case) can be described as the sum (a superposition) of an infinite number of individual periodic events (e.g. waves). This process is known as Fourier synthesis. The inverse process, the decomposition of the periodic event or object yields the individual components and is called Fourier analysis. How Fourier synthesis works in practice is shown in Figure 4. To keep the example most simple, we will first consider only the projection (a shadow image) of the black squares onto the horizontal a-axis in the beginning (Figure 3). 

Conversion of the as-deposited film into the crystalline state has been carried out by a variety of methods. The most typical approach is a two-step heat treatment process involving separate low-temperature pyrolysis ( 300 to 350°C) and high-temperature ( 550 to 750°C) crystallization anneals. The times and temperatures utilized depend upon precursor chemistry, film composition, and layer thickness. At the laboratory scale, the pyrolysis step is most often carried out by simply placing the film on a hot plate that has been preset to the desired temperature. Nearly always, pyrolysis conditions are chosen based on the thermal decomposition behavior of powders derived from the same solution chemistry. Thermal gravimetric analysis (TGA) is normally employed for these studies, and while this approach seems less than ideal, it has proved reasonably effective. A few investigators have studied organic pyrolysis in thin films by Fourier transform infrared spectroscopy (FTIR) using reflectance techniques. - This approach allows for an in situ determination of film pyrolysis behavior. 

Therefore we decided to gather new data about the laser-induced decomposition of polyimide and contrast them with results from pyrolysis using the same experimental technique, i.e., diffuse reflectance infrared Fourier-transform (DRIFT) spectroscopy. 

In the past, the large amount of spectral data generated by NIR instruments challenged the ability of computers to provide computations within a reasonable time frame. Mathematical techniques, therefore, which offered a reduction of the raw data, but with minimum loss of information, were often employed. The decomposition of 1000 spectral data points to 100 Fourier transform coefficients provides a great saving in computational time with very little loss of... 

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