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
Here,
is called the forward () Fourier transform,
and
is called the inverse () Fourier transform.
The notation is introduced in Trott (2004,
p. xxxiv), and and are sometimes
also used to denote the Fourier transform and inverse Fourier transform, respectively
(Krantz 1999, p. 202).
Note that 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
To restore the symmetry of the transforms, the convention
is sometimes used (Mathews and Walker 1970, p. 102).
In general, the Fourier transform pair may be defined using two arbitrary constants and as
The Fourier transform of a function
is implemented as FourierTransform[f, x, k], and different
choices of and can be used by
passing the optional FourierParameters-> a, b option. By default, Mathematica takes FourierParameters as
. Unfortunately, a number of other
conventions are in widespread use. For example, is used in
modern physics, is used in pure mathematics and
systems engineering, is used in probability theory for
the computation of the characteristic
function, is used in classical physics, and
is used in signal processing.
In this work, following Bracewell (1999, pp. 6-7), it is always assumed that
and unless
otherwise stated. This choice often results in greatly simplified transforms
of common functions such as 1, , etc.
Since any function can be split up into even and odd portions and ,
a Fourier transform can always be expressed in terms of the Fourier cosine transform and Fourier sine transform as
|
(19)
|
A function has a forward and inverse Fourier
transform such that
|
(20)
|
provided that
1. exists.
2. There are a finite number of discontinuities.
3. The function has bounded variation. A sufficient weaker condition is fulfillment of the Lipschitz
condition
(Ramirez 1985, p. 29). 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
Therefore,
The Fourier transform is also symmetric since
implies .
Let denote the convolution, then the transforms of convolutions of functions
have particularly nice 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
Fourier transform known as the Wiener-Khinchin
theorem. Let , and denote the complex conjugate of , then the Fourier
transform of the absolute square
of is given by
|
(33)
|
The Fourier transform of a derivative of a function is simply related
to the transform of the function itself. Consider
|
(34)
|
Now use integration by parts
|
(35)
|
with
and
then
|
(40)
|
The first term consists of an oscillating function times . But if the
function is bounded so that
|
(41)
|
(as any physically significant signal must be), then the term vanishes, leaving
This process can be iterated for the th derivative to yield
|
(44)
|
The important modulation theorem of Fourier transforms allows
to be expressed in terms of
as follows,
Since the derivative of the Fourier
transform is given by
|
(49)
|
it follows that
|
(50)
|
Iterating gives the general formula
The variance of a Fourier transform
is
|
(53)
|
and it is true that
|
(54)
|
If has the Fourier transform ,
then the Fourier transform has the shift property
so has the Fourier transform
|
(57)
|
If has a Fourier transform ,
then the Fourier transform obeys a similarity theorem.
|
(58)
|
so has the Fourier transform
|
(59)
|
The "equivalent width" of a Fourier transform is
The "autocorrelation width" is
where denotes the cross-correlation of and and is the complex conjugate.
Any operation on which leaves its area unchanged leaves unchanged,
since
|
(64)
|
The following table summarized some common Fourier transform pairs.
In two dimensions, the Fourier transform becomes
Similarly, the -dimensional Fourier transform can be
defined for , by
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, 3rd ed. New
York: McGraw-Hill, 1999.
Brigham, E. O. The Fast Fourier Transform and Applications. Englewood
Cliffs, NJ: Prentice Hall, 1988.
Folland, G. B. Real Analysis: Modern Techniques and their Applications, 2nd ed.
New York: Wiley, 1999.
James, J. F. A Student's Guide to Fourier Transforms with Applications in Physics
and Engineering. New York: Cambridge University Press, 1995.
Kammler, D. W. A First Course in Fourier Analysis. Upper Saddle River,
NJ: Prentice Hall, 2000.
Körner, T. W. Fourier Analysis. Cambridge, England: Cambridge University
Press, 1988.
Krantz, S. G. "The Fourier Transform." §15.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser,
pp. 202-212, 1999.
Mathews, J. and Walker, R. L. Mathematical Methods of Physics, 2nd ed. Reading, MA: W. A. Benjamin/Addison-Wesley,
1970.
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.
Oberhettinger, F. Fourier Transforms of Distributions and Their Inverses: A Collection
of Tables. New York: Academic Press, 1973.
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 FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press, 1989.
Ramirez, R. W. The FFT: Fundamentals and Concepts. Englewood Cliffs, NJ:
Prentice-Hall, 1985.
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.
Stein, E. M. and Weiss, G. L. Introduction to Fourier Analysis on Euclidean Spaces. Princeton,
NJ: Princeton University Press, 1971.
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.
Trott, M. The Mathematica GuideBook for Programming. New York:
Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.
Walker, J. S. Fast Fourier Transforms, 2nd ed. Boca Raton, FL: CRC Press,
1996.
Weisstein, E. W. "Books about Fourier Transforms." http://www.ericweisstein.com/encyclopedias/books/FourierTransforms.html.
|