| 释义 |
Fourier TransformThe Fourier transform is a generalization of the Complex Fourier Series in the limit as . Replace the discrete  with the continuous  while letting  . Then change the sum toan Integral, and the equations become
Here,
 | (3) |
 ) Fourier transform, and
 | (4) |
 ) Fourier transform. Some authors (especially physicists) prefer to write the transformin terms of angular frequency  instead of the oscillation frequency  . However, this destroysthe symmetry, resulting in the transform pair
In general, the Fourier transform pair may be defined using two arbitrary constants  and  as
The Mathematica program (Wolfram Research, Champaign, IL) calls the $FourierOverallConstant and the $FourierFrequencyConstant, and defines  by default. Morse and Feshbach(1953) use  and  . In this work, following Bracewell (1965, pp. 6-7),  and  unlessotherwise stated.
Since any function can be split up into Even and Odd portions  and ,
 | (9) |
 | (10) |
A function  has a forward and inverse Fourier transform such that
 | (11) |
- 1.
 exists. - 2. Any discontinuities are finite.
- 3. The function has bounded variation. A Sufficient weaker condition is fulfillment of the Lipschitz Condition.
The smoother a function (i.e., the larger the number of continuous Derivatives), the more compact itsFourier transform.
The Fourier transform is linear, since if and  have Fourier Transforms  and  , then  | |  | (12) |
Therefore,
 | (13) |
The Fourier transform is also symmetric since  implies  .
Let  denote the Convolution, then the transforms of convolutions of functions have particularlynice transforms,
The first of these is derived as follows:
where  .
There is also a somewhat surprising and extremely important relationship between the Autocorrelation and the Fouriertransform known as the Wiener-Khintchine Theorem. Let  , and  denote the ComplexConjugate of  , then the Fourier Transform of the Absolute Square of is given by
 | (19) |
The Fourier transform of a Derivative  of a function is simply related to the transform of the function itself. Consider
 | (20) |
 | (21) |
 | (22) |
 | (23) |
 | (24) |
 | (25) |
 | (26) |
 th Derivative to yield
 | (27) |
The important Modulation Theorem of Fourier transforms allows  to be expressed in termsof as follows,
Since the Derivative of the Fourier Transform is given by
 | (29) |
 | (30) |
 | (31) |
 | (32) |
 | (33) |
If has the Fourier Transform , then the Fourier transform has the shift property
so  has the Fourier Transform
 | (35) |
If has a Fourier Transform , then the Fourier transform obeys a similarity theorem.
 | (36) |
 has the Fourier Transform  .
The ``equivalent width'' of a Fourier transform is
 | (37) |
 | (38) |
 denotes the Cross-Correlation of  and  .
Any operation on which leaves its Area unchanged leaves  unchanged, since
 | (39) |
In 2-D, the Fourier transform becomes
 | (40) |
 | (41) |
 ,  by
See also Autocorrelation, Convolution, Discrete Fourier Transform, Fast Fourier Transform,Fourier Series, Fourier-Stieltjes Transform, Hankel Transform, Hartley Transform,Integral Transform, Laplace Transform, Structure Factor, Winograd Transform
References
 Fourier TransformsArfken, G. ``Development of the Fourier Integral,'' ``Fourier Transforms--Inversion Theorem,'' and ``Fourier Transform of Derivatives.'' §15.2-15.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 794-810, 1985. Blackman, R. B. and Tukey, J. W. The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, 1959. Bracewell, R. The Fourier Transform and Its Applications. New York: McGraw-Hill, 1965. Brigham, E. O. The Fast Fourier Transform and Applications. Englewood Cliffs, NJ: Prentice Hall, 1988. James, J. F. A Student's Guide to Fourier Transforms with Applications in Physics and Engineering. New York: Cambridge University Press, 1995. Körner, T. W. Fourier Analysis. Cambridge, England: Cambridge University Press, 1988. Morrison, N. Introduction to Fourier Analysis. New York: Wiley, 1994. Morse, P. M. and Feshbach, H. ``Fourier Transforms.'' §4.8 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 453-471, 1953. Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, 1989. Sansone, G. ``The Fourier Transform.'' §2.13 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 158-168, 1991. Sneddon, I. N. Fourier Transforms. New York: Dover, 1995. Sogge, C. D. Fourier Integrals in Classical Analysis. New York: Cambridge University Press, 1993. Spiegel, M. R. Theory and Problems of Fourier Analysis with Applications to Boundary Value Problems. New York: McGraw-Hill, 1974. Strichartz, R. Fourier Transforms and Distribution Theory. Boca Raton, FL: CRC Press, 1993. Titchmarsh, E. C. Introduction to the Theory of Fourier Integrals, 3rd ed. Oxford, England: Clarendon Press, 1948. Tolstov, G. P. Fourier Series. New York: Dover, 1976. Walker, J. S. Fast Fourier Transforms, 2nd ed. Boca Raton, FL: CRC Press, 1996.
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