Fourier spectrum integral method

The algorithms most fi equently used for calculation of fractal coefficients from the AFM results are [58] Fourier spectrum integral method, surface-perimeter method, structural function method and variable method. To determine the surface dimension by the Fourier spectrum integral method it is necessary to obtain the picture of the surface 2D FFT generating amplitude and time of the matrix (more detail s are given in paper [58]. Assuming the surface function as f(x,y), the Fourier transform in two-dimensional space can be expressed as [58] ... 

In principle, Mq can be determined through the first point in time domain for t = 0 or by integration of L. However, under experimental conditions distortions in the acquired time signal (e.g., due to eddy currents) are transferred to the frequency domain by Fourier transformation and can result in significant differences in quantification. A straightforward quantification method in the frequency domain is simply to determine the total integrated area under a resonance in a distinct frequency range of the spectrum. This method works well for spectra with well separated resonance lines and without... 

Fourier demonstrated that any periodic function, or wave, in any dimension, could always be reconstructed from an infinite series of simple sine waves consisting of integral multiples of the wave s own frequency, its spectrum. The trick is to know, or be able to find, the amplitude and phase of each of the sine wave components. Conversely, he showed that any periodic function could be decomposed into a spectrum of sine waves, each having a specific amplitude and phase. The former process has come to be known as a Fourier synthesis, and the latter as a Fourier analysis. The methods he proposed for doing this proved so powerful that he was rewarded by his mathematical colleagues with accusations of witchcraft. This reflects attitudes which once prevailed in academia, and often still do. 

Now, let us suppose that the system is in a black-box, and that all we can know of it is the observable Q, sampled up to a finite time interval. This is the typical situation occurring in Physics, where one obtains information on some system on the basis of the output of an experiment. The amplitudes and frequencies (6) can be numerically computed from g(t), for example, by means of the frequency analysis method (Laskar 1990, Laskar et al. 1992). However, if we are interested mainly in recognizing the quasi-periodic nature of the solution, it is not necessary to use a refined frequency analysis, but it is sufficient to compute the Fast Fourier transform of time interval [—T,T], where >( ) is a suitable analytic window on [—T, T] (see Section 4 for details). Figure 1 shows an example of such an analysis. Within the precision of our computation (a line is identified with an error of about 10-5 in frequency) we can easily recognize that the spectrum of g(t) is a line spectrum. Now, we consider the more interesting quasi-integrable Hamiltonian ... 

The most difficult problems relate to quantification, normally involving integrating individual peaks or spectra arising from specific compounds. Some methods for deconvolution distort these, for example, many methods for Fourier transform resolution enhancement are dependent on peakwidths of individual components. If a spectrum consists of peaks of different widths, each peak will be enhanced to a different extent, making relative integrals meaningless. Other approaches, such as in chemo-metric multivariate resolution, should preserve relative integrals. 

Four different methods used for integrated-path remote gas sensing are discussed here. One of these (tunable diode laser absorption spectroscopy, TOLAS) uses a narrow linewidth source of radiation (usually a laser diode) and the other three methods use broadband sources of radiation. These three analyze the spectrum of the radiation after it has traversed the atmospheric path in different ways both differential optical absorption spectroscopy (DOAS) and Fourier transform infrared (FTIR) spectroscopy analyze the entire spectrum over the spectral region of interest, whilst absorption correlation methods record the spectrum after it has been filtered optically with either an optical filter or a sample of the target gas itself. These four methods use an active source of radiation. It is also possible to carry out integrated-path remote gas sensing using a passive source. 

As for convex models, several useful methods can be used (Ben-Haim and Elishakoff 1990 Ben-Haim et al. 1996 Pantelides and Tzan 1996 Tzan and Pantelides 1996 Baratta et al. 1998). A convex model is defined mathematically as a set of functions. Each function is a realization of an uncertain event. Convex models for ground motion modeling depend on the level of prior information available. Examples are a local energy-bound convex model, an integral energy-bound convex model, an envelope-bound convex model, a Fourier-envelope convex model, and a response-spectrum-envelope convex model (Ben-Haim et al. 1996). One of the merits of the convex... 

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