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Fourier Transform

The Fourier transform is a generalization of the Complex Fourier Series in the limit as $L\to\infty$. Replace the discrete $A_n$ with the continuous $F(k)\,dk$ while letting $n/L\to k$. Then change the sum to an Integral, and the equations become

$\displaystyle f(x)$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty F(k)e^{2\pi ikx}\,dk$ (1)
$\displaystyle F(k)$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty f(x)e^{-2\pi ikx}\,dx.$ (2)

F(k)={\mathcal F}[f(x)]=\int_{-\infty}^\infty f(x)e^{-2\pi ikx}\,dx
\end{displaymath} (3)

is called the forward ($-i$) Fourier transform, and
f(x)={\mathcal F}^{-1}[F(k)]=\int_{-\infty}^\infty F(k)e^{2\pi ikx}\,dk
\end{displaymath} (4)

is called the inverse ($+i$) Fourier transform. Some authors (especially physicists) prefer to write the transform in terms of angular frequency $\omega\equiv 2\pi\nu$ instead of the oscillation frequency $\nu$. However, this destroys the symmetry, resulting in the transform pair
$\displaystyle H(\nu)$ $\textstyle =$ $\displaystyle {\mathcal F}[h(t)] = \int_{-\infty}^\infty h(t)e^{-i\omega t}\,dt$ (5)
$\displaystyle h(t)$ $\textstyle =$ $\displaystyle {\mathcal F}^{-1}[H(\nu)] = {1\over 2\pi}\int_{-\infty}^\infty H(\nu)e^{i\omega t}\,d\omega.$ (6)

In general, the Fourier transform pair may be defined using two arbitrary constants $A$ and $B$ as
$\displaystyle F(\omega)$ $\textstyle =$ $\displaystyle A\int_{-\infty}^\infty f(t)e^{Bi\omega t}\,dt$ (7)
$\displaystyle f(t)$ $\textstyle =$ $\displaystyle {B\over 2\pi A}\int_{-\infty}^\infty F(\omega)e^{-Bi\omega t}\,d\omega.$ (8)

The Mathematica ${}^{\scriptstyle\circledRsymbol}$ program (Wolfram Research, Champaign, IL) calls $A$ the $FourierOverallConstant and $B$ the $FourierFrequencyConstant, and defines $A=B=1$ by default. Morse and Feshbach (1953) use $B=1$ and $A=1/\sqrt{2\pi}$. In this work, following Bracewell (1965, pp. 6-7), $A=1$ and $B=-2\pi$ unless otherwise stated.

Since any function can be split up into Even and Odd portions $E(x)$ and $O(x)$,

f(x) = {\textstyle{1\over 2}}[f(x)+f(-x)]+ {\textstyle{1\over 2}}[f(x)-f(-x)] = E(x)+O(x),
\end{displaymath} (9)

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

{\mathcal F}[f(x)] = \int^\infty_{-\infty} E(x)\cos(2\pi kx)\,dx - i\int^\infty_{-\infty} O(x)\sin(2\pi kx)\,dx.
\end{displaymath} (10)

A function $f(x)$ has a forward and inverse Fourier transform such that

\int_{-\infty}^\infty e^{2\pi ikx}\left[{\int_...
\quad {\rm for\ } f(x) {\rm\ discontinuous\ at\ } x,\cr}
\end{displaymath} (11)

provided that
1. $\int_{-\infty}^\infty \vert f(x)\vert\,dx$ 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 $f(x)$ and $g(x)$ have Fourier Transforms $F(k)$ and $G(k)$, then

$\int [af(x)+bg(x)]e^{-2\pi ikx}\,dx &= a\int_{-\infty}^\infty f(x)e^{-2\pi ikx}\,dx+b\int_{-\infty}^\infty g(x)e^{-2\pi ikx}\,dx$
$ &= F(k)+G(k).&$ (12)

{\mathcal F}[af(x)+bg(x)] = a{\mathcal F}[f(x)]+b{\mathcal F}[g(x)]=aF(k)+bG(k).
\end{displaymath} (13)

The Fourier transform is also symmetric since $F(k)={\mathcal F}[f(x)]$ implies $F(-k)={\mathcal F}[f(x)]$.

Let $f*g$ denote the Convolution, then the transforms of convolutions of functions have particularly nice transforms,

$\displaystyle {\mathcal F}[f*g]$ $\textstyle =$ $\displaystyle {\mathcal F}[f]{\mathcal F}[g]$ (14)
$\displaystyle {\mathcal F}[fg]$ $\textstyle =$ $\displaystyle {\mathcal F}[f]*{\mathcal F}[g]$ (15)
$\displaystyle {\mathcal F}[{\mathcal F}(f){\mathcal F}(g)]$ $\textstyle =$ $\displaystyle f*g$ (16)
$\displaystyle {\mathcal F}[{\mathcal F}(f)*{\mathcal F}(g)]$ $\textstyle =$ $\displaystyle fg.$ (17)

The first of these is derived as follows:

$\displaystyle {\mathcal F}[f*g]$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-2\pi ikx}f(x')g(x-x')\,dx'\,dx$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty [e^{-2\pi ikx'}f(x')\,dx'][e^{-2\pi ik(x-x')}g(x-x')\,dx]$  
  $\textstyle =$ $\displaystyle \left[{\int_{-\infty}^\infty e^{-2\pi ikx'}f(x')\,dx'}\right]\left[{\int_{-\infty}^\infty e^{-2\pi ikx''}g(x'')\,dx''}\right]$  
  $\textstyle =$ $\displaystyle {\mathcal F}[f]{\mathcal F}[g],$ (18)

where $x''\equiv x-x'$.

There is also a somewhat surprising and extremely important relationship between the Autocorrelation and the Fourier transform known as the Wiener-Khintchine Theorem. Let ${\mathcal F}[f(x)] = F(k)$, and $F^*$ denote the Complex Conjugate of $F$, then the Fourier Transform of the Absolute Square of $F(k)$ is given by

{\mathcal F}[\vert F(k)\vert^2] = \int_{-\infty}^\infty f^*(\tau)f(\tau+x)\,d\tau.
\end{displaymath} (19)

The Fourier transform of a Derivative $f'(x)$ of a function $f(x)$ is simply related to the transform of the function $f(x)$ itself. Consider

{\mathcal F}[f'(x)] = \int_{-\infty}^\infty f'(x)e^{-2\pi ikx}\,dx.
\end{displaymath} (20)

Now use Integration by Parts
\int v\,du=[uv]-\int u\,dv
\end{displaymath} (21)

du=f'(x)\,dx \qquad v=e^{-2\pi ikx}
\end{displaymath} (22)

u=f(x) \qquad dv=-2\pi ike^{-2\pi ikx}\,dx,
\end{displaymath} (23)


{\mathcal F}[f'(x)] = [f(x)e^{-2\pi ikx}]^\infty_{-\infty} -\int_{-\infty}^\infty f(x)(-2\pi ike^{-2\pi ikx}\,dx).
\end{displaymath} (24)

The first term consists of an oscillating function times $f(x)$. But if the function is bounded so that
\lim_{x\to\pm\infty} f(x)=0
\end{displaymath} (25)

(as any physically significant signal must be), then the term vanishes, leaving
{\mathcal F}[f'(x)] = 2\pi ik \int_{-\infty}^\infty f(x)e^{-2\pi ikx}\,dx = 2\pi ik{\mathcal F}[f(x)].
\end{displaymath} (26)

This process can be iterated for the $n$th Derivative to yield
{\mathcal F}[f^{(n)}(x)] = (2\pi ik)^n{\mathcal F}[f(x)].
\end{displaymath} (27)

The important Modulation Theorem of Fourier transforms allows ${\mathcal F}[\cos(2\pi k_0x)f(x)]$ to be expressed in terms of ${\mathcal F}[f(x)] = F(k)$ as follows,

$\displaystyle {\mathcal F}[\cos(2\pi k_0x)f(x)]$ $\textstyle \equiv$ $\displaystyle \int_{-\infty}^\infty f(x)\cos(2\pi k_0x)e^{-2\pi ikx}\,dx$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\int_{-\infty}^\infty f(x)e^{2\pi ik_0x}e^{...
...\textstyle{1\over 2}}\int_{-\infty}^\infty f(x)e^{-2\pi ik_0x}e^{-2\pi ikx}\,dx$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\int_{-\infty}^\infty f(x)e^{-2\pi i(k-k_0)x}\,dx+{\textstyle{1\over 2}}\int_{-\infty}^\infty f(x)e^{-2\pi i(k+k_0)x}\,dx$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[F(k-k_0)+F(k+k_0)].$ (28)

Since the Derivative of the Fourier Transform is given by

F'(k)\equiv{d\over dk}{\mathcal F}[f(x)]=\int_{-\infty}^\infty (-2\pi ix)f(x)e^{-2\pi ikx}\,dx,
\end{displaymath} (29)

it follows that
F'(0)=-2\pi i\int_{-\infty}^\infty xf(x)\,dx.
\end{displaymath} (30)

Iterating gives the general Formula
\mu_n \equiv \int_{-\infty}^\infty x^nf(x)\,dx = {F^{(n)}(0)\over (-2\pi i)^n}.
\end{displaymath} (31)

The Variance of a Fourier Transform is
{\sigma_f}^2 = \left\langle{(xf-\left\langle{xf}\right\rangle{})^2}\right\rangle{},
\end{displaymath} (32)

and it is true that
\end{displaymath} (33)

If $f(x)$ has the Fourier Transform $F(k)$, then the Fourier transform has the shift property

$\displaystyle \int_{-\infty}^\infty f(x-x_0)e^{-2\pi ikx}\,dx$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty f(x-x_0)e^{-2\pi i(x-x_0)k} e^{-2\pi i(kx_0)}\,d(x-x_0)$  
  $\textstyle =$ $\displaystyle e^{-2\pi ikx_0}F(k),$ (34)

so $f(x-x_0)$ has the Fourier Transform
{\mathcal F}[f(x-x_0)]=e^{-2\pi ikx_0}F(k).
\end{displaymath} (35)

If $f(x)$ has a Fourier Transform $F(k)$, then the Fourier transform obeys a similarity theorem.

\int_{-\infty}^\infty f(ax)e^{-2\pi ikx}\,dx = {1\over \vert...
...k/a)}\,d(ax) = {1\over \vert a\vert} F\left({k\over a}\right),
\end{displaymath} (36)

so $f(ax)$ has the Fourier Transform $\vert a\vert^{-1}F\left({k\over a}\right)$.

The ``equivalent width'' of a Fourier transform is

w_e \equiv {\int_{-\infty}^\infty f(x)\,dx\over f(0)} = {F(0)\over\int_{-\infty}^\infty F(k)\,dk}.
\end{displaymath} (37)

The ``autocorrelation width'' is
w_a\equiv {\int_{-\infty}^\infty f\star f^*\,dx\over [f\star...
...{-\infty}^\infty f^*\,dx\over \int_{-\infty}^\infty ff^*\,dx},
\end{displaymath} (38)

where $f\star g$ denotes the Cross-Correlation of $f$ and $g$.

Any operation on $f(x)$ which leaves its Area unchanged leaves $F(0)$ unchanged, since

\int_{-\infty}^\infty f(x)\,dx ={\mathcal F}[f(0)] = F(0).
\end{displaymath} (39)

In 2-D, the Fourier transform becomes

F(x,y) = \int_{-\infty}^\infty \int_{-\infty}^\infty f(k_x,k_y)e^{-2\pi i(k_xx+k_yy)}\,dk_x\,dk_y
\end{displaymath} (40)

f(k_x,k_y) = \int_{-\infty}^\infty \int_{-\infty}^\infty F(x,y)e^{2\pi i(k_xx+k_yy)}\,dx\,dy.
\end{displaymath} (41)

Similarly, the $n$-D Fourier transform can be defined for ${\bf k}$, ${\bf x}\in\Bbb{R}^n$ by
$\displaystyle F({\bf x})$ $\textstyle =$ $\displaystyle \underbrace{\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty}_n
f({\bf k})e^{-2\pi i{\bf k}\cdot{\bf x}}\,d^n{\bf k}$ (42)
$\displaystyle f({\bf k})$ $\textstyle =$ $\displaystyle \underbrace{\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty}_n
F({\bf x})e^{2\pi i{\bf k}\cdot{\bf x}}\,d^n{\bf x}.$ (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


Fourier Transforms

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.

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© 1996-9 Eric W. Weisstein