Фуријеова трансформација

Фуријеова трансформација разлаже функцију времена (сигнал) у фреквенције које га чине, на сличан начин као што музички акорди могу бити изражени као фреквенције његових саставних нота.

Историја

Жозеф Фурије је 1822. године показао да неке функције могу бити записане као бесконачна сума хармоника.^[1]

Дефиниција

Фуријеова трансформација сигнала $f(t)$ рачуна се на следећи начин:

$F(j\omega )=\int \limits _{-\infty }^{\infty }f(t)e^{-j\omega t}dt$

$F(j\omega )$ је комплексна величина. Њен модуо назива се спектрална густина амплитуда, а аргумент спектрална густина фаза.^[2]^[3]

Инверзија

Инверзна Фуријеова трансформација је:

$f(t)={\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }F(j\omega )e^{j\omega t}d\omega$

Особине Фуријеове трансформације

Линеарност

За било које комплексне бројеве $a$ и $b$ , ако је $h(x)=af(x)+bg(x)$ , важи да је $H(j\omega )=aF(j\omega )+bG(j\omega )$ .

Транслација

За било који реалан број $x_{0}$ , ако је $h(x)=f(x-x_{0})$ , важи да је $H(j\omega )=e^{j\omega tx_{0}}F(j\omega )$ .

Види још

Референце

^ Fourier, Jean Baptiste Joseph baron (1822). Théorie analytique de la chaleur (на језику: француски). Chez Firmin Didot, père et fils.
^ Kaiser 1994, стр. 29.
^ Rahman 2011, стр. 11.

Литература

Bailey, David H.; Swarztrauber, Paul N. (1994), „A fast method for the numerical evaluation of continuous Fourier and Laplace transforms” (PDF), SIAM Journal on Scientific Computing, 15 (5): 1105—1110, CiteSeerX 10.1.1.127.1534  , doi:10.1137/0915067, Архивирано из оригинала (PDF) 20. 07. 2008. г., Приступљено 18. 10. 2020 .

Boashash, B., ур. (2003), Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Oxford: Elsevier Science, ISBN 978-0-08-044335-5 .

Bochner, S.; Chandrasekharan, K. (1949), Fourier Transforms, Princeton University Press .

Bracewell, R. N. (2000), The Fourier Transform and Its Applications (3rd изд.), Boston: McGraw-Hill, ISBN 978-0-07-116043-8 .

Campbell, George; Foster, Ronald (1948), Fourier Integrals for Practical Applications, New York: D. Van Nostrand Company, Inc. .

Champeney, D.C. (1987), A Handbook of Fourier Theorems, Cambridge University Press .

Chatfield, Chris (2004), The Analysis of Time Series: An Introduction, Texts in Statistical Science (6th изд.), London: Chapman & Hall/CRC .

Clozel, Laurent; Delorme, Patrice (1985), „Sur le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs réels”, Comptes Rendus de l'Académie des Sciences, Série I, 300: 331—333 .

Condon, E. U. (1937), „Immersion of the Fourier transform in a continuous group of functional transformations”, Proc. Natl. Acad. Sci., 23 (3): 158—164, Bibcode:1937PNAS...23..158C, PMC 1076889  , PMID 16588141, doi:10.1073/pnas.23.3.158 .

de Groot, Sybren R.; Mazur, Peter (1984), Non-Equilibrium Thermodynamics (2nd изд.), New York: Dover .

Duoandikoetxea, Javier (2001), Fourier Analysis, American Mathematical Society, ISBN 978-0-8218-2172-5 .

Dym, H.; McKean, H. (1985), Fourier Series and Integrals, Academic Press, ISBN 978-0-12-226451-1 .

Erdélyi, Arthur, ур. (1954), Tables of Integral Transforms, Vol. 1, McGraw-Hill .

Feller, William (1971), An Introduction to Probability Theory and Its Applications, Vol. II (2nd изд.), New York: Wiley, MR 0270403 .

Folland, Gerald (1989), Harmonic analysis in phase space, Princeton University Press .

Fourier, J.B. Joseph (1822), Théorie analytique de la chaleur (на језику: French), Paris: Firmin Didot, père et fils, OCLC 2688081 .

Fourier, J.B. Joseph (1878) [1822], The Analytical Theory of Heat, Превод: Alexander Freeman, The University Press (translated from French).

Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015), Zwillinger, Daniel; Moll, Victor Hugo, ур., Table of Integrals, Series, and Products (на језику: енглески), Превод: Scripta Technica, Inc. (8th изд.), Academic Press, ISBN 978-0-12-384933-5 .

Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Prentice-Hall, ISBN 978-0-13-035399-3 .

Grafakos, Loukas; Teschl, Gerald (2013), „On Fourier transforms of radial functions and distributions”, J. Fourier Anal. Appl., 19: 167—179, arXiv:1112.5469  , doi:10.1007/s00041-012-9242-5 .

Greiner, W.; Reinhardt, J. (1996), Field Quantization , Springer, ISBN 978-3-540-59179-5 .

Gelfand, I.M.; Shilov, G.E. (1964), Generalized Functions, Vol. 1, New York: Academic Press (translated from Russian).

Gelfand, I.M.; Vilenkin, N.Y. (1964), Generalized Functions, Vol. 4, New York: Academic Press (translated from Russian).

Hewitt, Edwin; Ross, Kenneth A. (1970), Abstract harmonic analysis, Die Grundlehren der mathematischen Wissenschaften, Band 152, Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Springer, MR 0262773 .

Halidias, Nikolaos (2018), A generalisation of Laplace and Fourier transforms, Asian Journal of Mathematics and Computer Research .
Hörmander, L. (1976), Linear Partial Differential Operators, Vol. 1, Springer, ISBN 978-3-540-00662-6 .

Howe, Roger (1980), „On the role of the Heisenberg group in harmonic analysis”, Bulletin of the American Mathematical Society, 3 (2): 821—844, MR 578375, doi:10.1090/S0273-0979-1980-14825-9  .

James, J.F. (2011), A Student's Guide to Fourier Transforms (3rd изд.), Cambridge University Press, ISBN 978-0-521-17683-5 .

Jordan, Camille (1883), Cours d'Analyse de l'École Polytechnique, Vol. II, Calcul Intégral: Intégrales définies et indéfinies (2nd изд.), Paris .

Kaiser, Gerald (1994), „A Friendly Guide to Wavelets”, Physics Today, 48 (7): 57—58, Bibcode:1995PhT....48g..57K, ISBN 978-0-8176-3711-8, doi:10.1063/1.2808105 .

Kammler, David (2000), A First Course in Fourier Analysis, Prentice Hall, ISBN 978-0-13-578782-3 .

Katznelson, Yitzhak (1976), An Introduction to Harmonic Analysis, Dover, ISBN 978-0-486-63331-2 .

Kirillov, Alexandre; Gvishiani, Alexei D. (1982) [1979], Theorems and Problems in Functional Analysis, Springer (translated from Russian).

Knapp, Anthony W. (2001), Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, ISBN 978-0-691-09089-4 .

Kolmogorov, Andrey Nikolaevich; Fomin, Sergei Vasilyevich (1999) [1957], Elements of the Theory of Functions and Functional Analysis, Dover (translated from Russian).

Lado, F. (1971), „Numerical Fourier transforms in one, two, and three dimensions for liquid state calculations”, Journal of Computational Physics, 8 (3): 417—433, Bibcode:1971JCoPh...8..417L, doi:10.1016/0021-9991(71)90021-0 .

Müller, Meinard (2015), The Fourier Transform in a Nutshell. (PDF), In Fundamentals of Music Processing, Section 2.1, pages 40-56: Springer, ISBN 978-3-319-21944-8, doi:10.1007/978-3-319-21945-5, Архивирано из оригинала (PDF) 08. 04. 2016. г., Приступљено 18. 10. 2020 .

Paley, R.E.A.C.; Wiener, Norbert (1934), Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications (19), Providence, Rhode Island: American Mathematical Society .

Pinsky, Mark (2002), Introduction to Fourier Analysis and Wavelets, Brooks/Cole, ISBN 978-0-534-37660-4 .

Poincaré, Henri (1895), Théorie analytique de la propagation de la chaleur, Paris: Carré .

Polyanin, A. D.; Manzhirov, A. V. (1998), Handbook of Integral Equations, Boca Raton: CRC Press, ISBN 978-0-8493-2876-3 .

Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1992), Numerical Recipes in C: The Art of Scientific Computing, Second Edition (2nd изд.), Cambridge University Press .

Rahman, Matiur (2011), Applications of Fourier Transforms to Generalized Functions, WIT Press, ISBN 978-1-84564-564-9 .

Rudin, Walter (1987), Real and Complex Analysis (3rd изд.), Singapore: McGraw Hill, ISBN 978-0-07-100276-9 .

Simonen, P.; Olkkonen, H. (1985), „Fast method for computing the Fourier integral transform via Simpson's numerical integration”, Journal of Biomedical Engineering, 7 (4): 337—340, doi:10.1016/0141-5425(85)90067-6 .

Stein, Elias; Shakarchi, Rami (2003), Fourier Analysis: An introduction, Princeton University Press, ISBN 978-0-691-11384-5 .

Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9 .

Taneja, H.C. (2008), „Chapter 18: Fourier integrals and Fourier transforms”, Advanced Engineering Mathematics, Vol. 2, New Delhi, India: I. K. International Pvt Ltd, ISBN 978-8189866563 .

Titchmarsh, E. (1986) [1948], Introduction to the theory of Fourier integrals (2nd изд.), Oxford University: Clarendon Press, ISBN 978-0-8284-0324-5 .

Vretblad, Anders (2000), Fourier Analysis and its Applications, Graduate Texts in Mathematics, 223, New York: Springer, ISBN 978-0-387-00836-3 .

Whittaker, E. T.; Watson, G. N. (1927), A Course of Modern Analysis (4th изд.), Cambridge University Press .

Widder, David Vernon; Wiener, Norbert (август 1938), „Remarks on the Classical Inversion Formula for the Laplace Integral”, Bulletin of the American Mathematical Society, 44 (8): 573—575, doi:10.1090/s0002-9904-1938-06812-7 .

Wiener, Norbert (1949), Extrapolation, Interpolation, and Smoothing of Stationary Time Series With Engineering Applications, Cambridge, Mass.: Technology Press and John Wiley & Sons and Chapman & Hall .

Wilson, R. G. (1995), Fourier Series and Optical Transform Techniques in Contemporary Optics, New York: Wiley, ISBN 978-0-471-30357-2 .

Yosida, K. (1968), Functional Analysis, Springer, ISBN 978-3-540-58654-8 .

Спољашње везе

Encyclopedia of Mathematics
Weisstein, Eric W. „Fourier Transform”. MathWorld.

[1] Fourier, Jean Baptiste Joseph baron (1822). Théorie analytique de la chaleur (на језику: француски). Chez Firmin Didot, père et fils.

[2] Kaiser 1994, стр. 29.

[3] Rahman 2011, стр. 11.

[1]

[2]

[3]