Fourier analysis

Fourier analysis treats the representation of periodic functions as hnear combinations of sine and cosine basis functions. In chemical engineering, Fourier analysis is applied to study time-dependent signals in spectroscopy and to analyze the spatial structure of materials from scattering experiments. Here, the basic foundation of Fourier analysis is presented, with an emphasis upon implementation in MATLAB. 

Kielkopf [KIE 73] showed that the Fourier coefficients of a Voigt function are written as follows  

On the other hand, we will assume that the effects of both size and lattice distortions are described by a Voigt functiorr, meaning that the resulting function will also be a Voigt function. We can then write  

From these relatiorts and by applying eqiration [6.53], we get the mean crystal size  

We showed that, if we assumed a Gaussian microstrain distribution, then the Fourier coefficients related to these microstrains can be written  

This means that Voigt function fitting makes it possible to determine the size and the microstrain rate of each family of planes, along the direction perpendicular to this family of planes. 

What if we know the equation for the periodic signal x t) and wish to find the complex amplitudes ak for each harmonic The Fourier analysis equation gives this  

It should be clear, that integrating a sinusoid over a single period will give 0 for all sinusoids  

This holds for any harmonic also - while we have more positive and negative areas, as we have an equal munber of them then the sum will always be zero. In exponential form, we can therefore state  

Now consider the calculation of the inner product of two harmonically related sinusoids, where one is the complex conjugate of the other 

When k l, this quantity / is an other integer, which will represent one of the harmonics. But we know from equation 10.1.4 that this will always evaluate to 0. However, when k = I, then — / = 0 so the integral is 

An alternative numerical method for resolving complicated signals is to analyze the frequency spectrum of the curves. The so-called Fourier transform analysis (FT analysis) approximates the sum of sine and cosine functions for empirically generated signals. First, 2 m points of supports, spaced equidistantly, are chosen (Fig. 2-2). We recommend choosing a multiple of four for 2 m and using the values 12, 24, 36, 72,. .. 

Then one can take advantage of the symmetry of the sine and cosine function. The most complex operation in the Fourier expansion, 

These algorithms require an enormous database and take relatively long to compute. Smoothing effects can be obtained when the higher frequencies are neglected. Further information is available in special mathematical textbooks and in [10, 25-27]. 

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Fourier transform is widely used for signal analysis purposes and is satisfactory when applied to signals where stationary features are of particular interest. However, it turns out to be very poor when dealing with defect detection, where it is the non stationary characteristics of the signal which has to be highlighted. The main reason is that in the Fourier analysis, the time parameter is discarded. 

The evaluation of the deconvolution results show that time resolution is better or equal to 1 with the chosen processing time unit of 0.08 microseconds (respectively a rate of 12.5 MHz). First signals processed conservatively have been acquired with a samplerate of 12.5 MHz. A Fourier analysis shows that the signals spectras do not have energy above 2.0 MHz. This means that a sampling rate of 4.0 MHz would have done the job as well. Due to the time base of the ADC an experimental check with a sample rate of 5.25 MHz has been carried out successfully. 

The most important feature of the Fourier analysis is the reduction of the multicoUi-nearity and ike dimension of ike original specira. However, ihe Fourier coefficients hear no. simple relationship to individual features of the spectrum so that it will not he clear what information is being used in calibration."... 

R. Vichnevetsky and J.B. Bowles, Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM Studies in Applied Mathematics, 1982. 

Time domains and frequeney domains are related through Fourier series and Fourier transforms. By Fourier analysis, a variable expressed as a funetion of time may be deeomposed into a series of oseillatory funetions (eaeh with a eharaeteristie frequeney), whieh when superpositioned or summed at eaeh time, will equal the original expression of the variable. This... 

Moews, P.C., Kretsinger, R.H. Refinement of the structure of carp muscle calcium-binding parvalbumin by model building and difference Fourier analysis. 

Whitney s Fourier Analysis Ashton s Rayleigh-Ritz Analysis... 

Hooke, C. J. and Venner, C. H., Surface Roughness Attenuation in Line and Point Contacts," Proc. Inst. Mech. Eng., PartJ J. Eng. Tribol., Vol. 214,2000, pp. 439-444. Morales-Espejel, G. E., Venner, C. H., and Greenwood, J. A., Kinematics of Transverse Real Roughness in Elastohydrody-namically Lubricated Line Contacts Using Fourier Analysis," Proc. Inst. Mech. Eng., PartJ.J. Eng. Tribol.,Vol.21A, No. J6,2000, pp. 523-534. 

Although other descriptions are possible, the mathematical concept that matches more closely the intuitive notion of smoothness is the frequency content of the function. Smooth functions are sluggish and coarse and characterized by very gradual changes on the value of the output as we scan the input space. This, in a Fourier analysis of the function, corresponds to high content of low frequencies. Furthermore, we expect the frequency content of the approximating function to vary with the position in the input space. Many functions contain high-frequency features dispersed in the input space that are very important to capture. The tool used to describe the function will have to support local features of multiple resolutions (variable frequencies) within the input space. 

The essence of analyzing an EXAFS spectrum is to recognize all sine contributions in x(k)- The obvious mathematical tool with which to achieve this is Fourier analysis. The argument of each sine contribution in Eq. (8) depends on k (which is known), on r (to be determined), and on the phase shift

characteristic property of the scattering atom in a certain environment, and is best derived from the EXAFS spectrum of a reference compound for which all distances are known. The EXAFS information becomes accessible, if we convert it into a radial distribution function, 0 (r), by means of Fourier transformation ... 

Figure 4.11. Left Simulated EXAFS spectrum of a dimer such as Cu2, showing that the EXAFS signal is the product of a sine function and a backscattering amplitude F(k) divided by k, as expressed by Eq. (6). Note that F k)/k remains visible as the envelope around the EXAFS signal xW- Right The Cu EXAFS spectrum of a cluster such as CU2O is the sum of a Cu-Cu and a Cu-O contribution. Fourier analysis is the mathematical tool used to...

Instead of this methodology, we have chosen to use Fourier analysis of the entire peak shape. By this procedure all of the above problems are eliminated. In particular, we focus on the cosine coefficients of the Fourier series representing a peak. The instrumental effects are readily removed, and the remaining coefficient of harmonic number, (n), A, can be written as a product ... 

As already noted in the case of the Scherrer equation, if we have a polydispersed ensemble of spheres the dimensions obtained by Fourier analysis correspond to ... 

A special type of data pre-treatment is the transformation of data into a smaller number of new variables. Principal components analysis is a natural example and we have treated it in Section 36.2.3 as PCR. Another way to summarize a spectrum in a few terms is through Fourier analysis. McClure [29] has shown how a NIR... 

W.F. McClure, A. Hamid, F.G. Giesbrecht and W.W, Weeks, Fourier analysis enhances NIR diffuse reflectance spectroscopy. Appl. Spectrosc., 38 (1988) 322-329. 

F. Dondi, A. Betti, L. Pasti, M.C. Pietrogrande and A. Felinger, Fourier analysis of multicomponent chromatograms — application to experimental chromatograms. Anal. Chem., 65 (1993) 2209-2222. 

The EEG is analyzed by Fourier analysis power density spectra are computed for periods of 4 seconds, segmented into six frequency bands, and averaged on each channel over timeblocks of 15 minutes. 

Fig. 3.47 is comparable to Fig. 3.41 for sinusoidal ac polarography if the tilted shape provides a net compensation of the charging current one obtains a symmetric bell-shaped curve of I in the square-wave polarogram, similar to that depicted in Fig. 3.42. In fact, virtually all of the statements made before on the sinusoidal technique are valid for the square-wave mode except for the rigid shape of its wave this conclusion is according to expectation, especially as Fourier analysis reveals the square wave to be a summation of a series of only...

Pattison, P. and Williams, B. (1976) Fermi surface parameters from fourier analysis of Compton profiles, Solid State Commun., 20, 585-588. 

Spiegel, Murray R., Schaum s Outline of Theory and Problems of Fourier Analysis, McGraw-Hill Book Company, New York (1974). 

Multidimensional image information can be processed in the same way as signal functions in general. In many cases, the basis of image processing is the two-dimensional Fourier analysis... 

This book is organized into five sections (1) Theory, (2) Columns, Instrumentation, and Methods, (3) Life Science Applications, (4) Multidimensional Separations Using Capillary Electrophoresis, and (5) Industrial Applications. The first section covers theoretical topics including a theory overview chapter (Chapter 2), which deals with peak capacity, resolution, sampling, peak overlap, and other issues that have evolved the present level of understanding of multidimensional separation science. Two issues, however, are presented in more detail, and these are the effects of correlation on peak capacity (Chapter 3) and the use of sophisticated Fourier analysis methods for component estimation (Chapter 4). Chapter 11 also discusses a new approach to evaluating correlation and peak capacity. 

The statistical model of peak overlap clearly explains that the number of observed peaks is much smaller than the number of components present in the sample. The Fourier analysis of multicomponent chromatograms can not only identify the ordered or disordered retention pattern but also estimate the average spot size, the number of detectable components present in the sample, the spot capacity, and the saturation factor (Felinger et al., 1990). Fourier analysis has been applied to estimate the number of detectable components in several complex mixtures. 

An adaptation of Fourier analysis to 2D separations can be established by calculating the autocovariance function (Marchetti et al., 2004). The theoretical background of that approach is that the power spectrum and the autocovariance function of a signal constitute a Fourier pair, that is, the power spectmm is obtained as the Fourier transform of the autocovariance function. 

Felinger, A., Pasti, L., Dondi, F. (1990). Fourier analysis of multicomponent chromatograms. Theory and models. Anal. Chem. 62, 1846. 

Presented in this manner, the analysis may proceed similarly to the treatment obtained from the Fourier analysis. C is the zero frequency component of the fit and A and B may be treated as the real and imaginary parts of the complex number. 

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