Thompson - Computing for Scientists and Engineers (523188), страница 68
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(An example is shown in Figure 9.19.) Describe qualitatively how and why the agreement between filtered and original data changeswith maxf. For what range of maxf values do you judge that you are probablyreproducing the EEG signal, as opposed to the EEG data, which contain somenoise? n374DISCRETE FOURIER TRANSFORMS AND FOURIER SERIESIf you followed the discussion in Section 9.6, you will object to this kind of filtering, because we have introduced a discontinuity into the Fourier amplitudes, ck,by suddenly (at maxf ) effectively turning them to zeros. When we Fourier transform these modified amplitudes we will get a Wilbraham-Gibbs overshoot introduced into the reconstructed EEG.
With such a bumpy function as our data represent and with such coarse time steps, this will not be obvious by looking at the results, but we know that the overshoot has to be there.The Lanczos damping factors provide a smooth transition from complete inclusion of the lowest frequencies in the filtered reconstruction to suppression (by a factor of about 0.64) of the highest frequency.Exercise 9.40Show analytically that integration of a continuous function over the rangeabout each point x of the function is equivalent to multiplying its k thfrequency component amplitude, ck, by the damping factor(9.96)These factors are called Lanczos damping factors. nThe Lanczos damping factors,are shown in Figure 9.21. Note that they are independent of the data being analyzed.
In our analysis, since the major amplitudesare concentrated around 8 Hz (Figure 9.19) suppressing higher-frequency amplitudes has relatively little effect on the reconstructed amplitudes. That is why in Figure 9.19 the Lanczos-filtered and original EEG data nearly coincide. Try for yourself the effects of Lanczos damping on the reconstruction of the EEG.FIGURE 9.21 EEG amplitudes (with sign) as a function of frequency, k, for the EEG data setV1 (from Table 9.1) are shown by shaded bars. The Lanczos-filter function,shown by the hollow bars.from (9.96) isREFERENCES ON FOURIER EXPANSIONS375Exercise 9.41Explore the effects of Lanczos damping in filtering Fourier expansions by usingthe program EEG FFT Analysis as follows:(a) Use filtering option 2 (Lanczos filtering) applied to one of the EEG data setsgiven in Table 9.1.
(By now, you probably have typed in these three data sets.)Make a plot comparing the original and filtered EEG. In this case, because filtering effects are likely to be small, it is a good idea to plot the difference (including sign) between the original and filtered values.(b) Modify the program slightly so that you can input your own choice of Fourier amplitudes, ck. Then make a data set such that the highest frequencies areemphasized. From this set of amplitudes reconstruct the original function byusing the inverse FFT after the program has Lanczos-filtered the amplitudes.Now you should notice a much larger effect from Lanczos filtering than in theEEG data. nFiltering is further considered in the texts on signal processing by Hamming, byOppenheim and Schafer, and by Embree and Kimble.
The last book has many algorithms, plus programs written in C, for digital signal processing. A suite of programs, primarily for FFT calculations, is provided in Chapter 12 of the numericalrecipes book by Press et al. Bracewell’s text on Fourier transforms presents manyengineering examples.REFERENCES ON FOURIER EXPANSIONSAbramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions,Dover, New York, 1964.Bôcher, M., “Introduction to the Theory of Fourier’s Series,” Annals of Mathematics, 7, 81, Sect. 9 (1906).Bracewell, R.
N., The Fourier Transform and Its Applications, McGraw-Hill,New York, second edition, 1986.Bracewell, R. N., The Hartley Transform, Oxford University Press, Oxford,England, 1986.Brigham, E. O., The Fast Fourier Transform and Its Applications, Prentice Hall,Englewood Cliffs, New Jersey, 1988.Cameron, J. R., and J. G. Skofronick, Medical Physics, Wiley, New York, 1978.Champeney, D.
C., A Handbook of Fourier Theorems, Cambridge UniversityPress, Cambridge, England, 1987.Cromwell, L., F. J. Weibell, and E. A. Pfeiffer, Biomedical Instrumentation andMeasurements, Prentice Hall, Englewood Cliffs, New Jersey, 1980.Embree, P. M., and B. Kimble, C Language Algorithms for Digital SignalProcessing, Prentice Hall, Englewood Cliffs, New Jersey, 1991.376DISCRETE FOURIER TRANSFORMS AND FOURIER SERIESGibbs, J. W., “Fourier’s Series,” Nature, 59, 200 (1898); erratum, Nature, 59,606 (1899): reprinted in The Collected Works of J. Willard Gibbs, LaymansGreen, New York, Vol. II, Part 2, p. 258, 1931.Hamming, R.
W., Digital Filters, Prentice Hall, Englewood Cliffs, New Jersey,third edition, 1989.Körner, T. W., Fourier Analysis, Cambridge University Press, Cambridge,England, 1988.Oppenheim, A. V., and R. W. Schafer, Discrete-Time Signal Processing, PrenticeHall, Englewood Cliffs, New Jersey, 1989.Peters, R. D., “Fourier Transform Construction by Vector Graphics,” AmericanJournal of Physics, 60, 439 (1992).Pratt, W.
K., Digital Image Processing, Wiley, New York, 1978.Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, NumericalRecipes in C, Cambridge University Press, New York, 1988.Protter, M. H., and C. B. Morrey, Intermediate Calculus, Springer-Verlag, NewYork, second edition, 1985.Roberts, R. A., and C. T. Mullis, Digital Signal Processing, Addison-Wesley,Reading, Massachusetts, 1987.Spehlmann, R., EEG Primer, Elsevier, New York, 1981.Thompson, W.
J., “Fourier Series and the Gibbs Phenomenon,” American Journalof Physics, 60, 425 (1992).Weaver, H. J., Applications of Discrete and Continuous Fourier Analysis, WileyInterscience, New York, 1983.Wilbraham, H., “On a Certain Periodic Function,” Cambridge and Dublin Mathematics Journal, 3, 198 (1848).Previous Home NextChapter 10FOURIER INTEGRAL TRANSFORMSIn this chapter we continue the saga of Fourier expansions that we began in Chapter 9 by exploring the discrete Fourier transform, the Fourier series, and the FastFourier Transform (FFT) algorithm. A major goal in this chapter is to extend theFourier series to the Fourier integral transform, thus completing our treatment ofFourier expansions that was outlined in Section 9.1.The outline of this chapter is as follows.
In Section 10.1 we make the transitionfrom Fourier series to Fourier integrals, then in Section 10.2 we give several examples of these transforms that are interesting for practical applications, especially theirapplication to Lorentzian and Gaussian functions. By Section 10.3 we have enoughanalytical preparation that we can start to emphasis applications of Fourier integraltransforms and their numerical approximations, so we investigate convolutions calculated from Fourier integral transforms, including the Voigt function that is usedextensively in analyzing optical spectra.
In Project 10 (Section 10.4) we develop aprogram for calculating convolutions by using the FFT, then apply it to calculate theline profile of a stellar spectrum. References on Fourier integral transforms roundout the chapter.10.1 FROM FOURIER SERIES TO FOURIER INTEGRALSIn the discussion of discrete Fourier transforms and Fourier series in Chapter 9, wefound that they predict a periodicity of any function expressed in terms of these twoexpansions. Suppose, however, that we want to describe an impressed force, a voltage pattern, or an image, that is not periodic. Can the Fourier series be adapted tothis use? It can, as we show in the following.377378FOURIER INTEGRAL TRANSFORMSThe transition from series to integralsOne way to make the transition from Fourier series to Fourier integrals is to allowthe upper and lower limits of the interval for a Fourier series to lie far outside therange of x values for which we will use the series.
The periodicity of the Fourierseries will then be inconsequential. This idea also leads to the discrete hamonics, k,in the Fourier series becoming continuous variables, thus establishing symmetry oftreatment with the x variables, as outlined in Section 9.1.Our method of deriving the Fourier integral transform is to set x- = -L andx+ = L, so that the interval x+- x- = 2L in the Fourier series for arbitrary intervals (Section 9.4).
Eventually we letto produce the integral transformfrom the series. In more detail, we temporarily setin (9.46), so thatunit step of k produces the changeThe Fourier series expansion of afunction y (x ), equation (9.48), then becomes(10.1)in which the Fourier coefficientsare given by(10.2)In these two expressions, in order to achieve symmetry we have split the factors inthe denominators as shown.
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