## Fourier Transform

The 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 to an Integral, and the equations become

 (1) (2)

Here,
 (3)

is called the forward () Fourier transform, and
 (4)

is called the inverse () Fourier transform. Some authors (especially physicists) prefer to write the transform in terms of angular frequency instead of the oscillation frequency . However, this destroys the symmetry, resulting in the transform pair
 (5) (6)

In general, the Fourier transform pair may be defined using two arbitrary constants and as
 (7) (8)

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 unless otherwise stated.

Since any function can be split up into Even and Odd portions and ,

 (9)

a Fourier transform can always be expressed in terms of the Fourier Cosine Transform and Fourier Sine Transform as

 (10)

A function has a forward and inverse Fourier transform such that

 (11)

provided that
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 its Fourier 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 particularly nice transforms,

 (14) (15) (16) (17)

The first of these is derived as follows:

 (18)

where .

There is also a somewhat surprising and extremely important relationship between the Autocorrelation and the Fourier transform known as the Wiener-Khintchine Theorem. Let , and denote the Complex Conjugate 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)

Now use Integration by Parts
 (21)

with
 (22)

 (23)

then

 (24)

The first term consists of an oscillating function times . But if the function is bounded so that
 (25)

(as any physically significant signal must be), then the term vanishes, leaving
 (26)

This process can be iterated for the th Derivative to yield
 (27)

The important Modulation Theorem of Fourier transforms allows to be expressed in terms of as follows,

 (28)

Since the Derivative of the Fourier Transform is given by

 (29)

it follows that
 (30)

Iterating gives the general Formula
 (31)

The Variance of a Fourier Transform is
 (32)

and it is true that
 (33)

If has the Fourier Transform , then the Fourier transform has the shift property

 (34)

so has the Fourier Transform
 (35)

If has a Fourier Transform , then the Fourier transform obeys a similarity theorem.

 (36)

so has the Fourier Transform .

The equivalent width'' of a Fourier transform is

 (37)

The autocorrelation width'' is
 (38)

where 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)

Similarly, the -D Fourier transform can be defined for , by
 (42) (43)

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

Arfken, 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.

© 1996-9 Eric W. Weisstein
1999-05-26