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Properties of Fourier Transforms

Dirac Delta Functions

Because the inverse transform of a transform returns the original function, this allows a definition of an interesting function called the Dirac delta function $\delta(x - x_o)$ . Combining the two equations in Eq. 18-5 into a single equation, and then interchanging the order of integration:

$$\begin{split}f(x) = & \frac{1}{2 \pi} \int_{-\infty}^{\infty}... ...nfty} e^{\imath k (x - \xi)} d k \right\} d\xi \ \end{split}$$ (18-8)

Apparently, a function can be defined

$$\delta(x - x_o) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{\imath k (x - \xi)} dk$$ (18-9)

that has the property

$$f(x_o) = \int_{-\infty}^{\infty} \delta(x-x_o) f(x) dx$$ (18-10)

in other words, $\delta$ picks out the value at $x = x_o$ and returns it outside of the integration.

Parseval's Theorem

The delta function can be used to derive an important conservation theorem.

If $f(x)$ represents the density of some function (i.e., a wave-function like $\psi(x)$ ), the square-magnitude of $f$ integrated over all of space should be the total amount of material in space.

$$\int_{-\infty}^{\infty} f(x) \bar{f}(x) dx = \int_{-\infty}^{\inf... ...\sqrt{2 \pi}} \bar{g}(\kappa) e^{-\imath \kappa x} d \kappa \right) \right\} dx$$ (18-11)

where the complex-conjugate is indicated by the over-bar. This exponentials can be collected together and the definition of the $\delta$ -function can be applied and the following simple result can is obtained

$$\int_{-\infty}^{\infty} f(x) \bar{f}(x) dx = \int_{-\infty}^{\infty} g(k) \bar{g}(k) dk =$$ (18-12)

which is Parseval's theorem. It says, that the magnitude of the wave-function, whether it is summed over real space or over momentum space must be the same.

Convolution Theorem

The convolution of two functions is given by

$$F(x) = p_1 (x) \star p_2 (x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} p_1(\eta) p_2(x - \eta) d \eta$$ (18-13)

If $p_1$ and $p_2$ can be interpreted as densities in probability, then this convolution quantity can be interpreted as ``the total joint probability due to two probability distributions whose arguments add up to $x$ .''¹

The proof is straightforward that the convolution of two functions, $p_1(x)$ and $p_2(x)$ , is a Fourier integral over the product of their Fourier transforms, $\psi_1(k)$ and $\psi_2(k)$ :

$$p_1 (x) \star p_2 (x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\i... ...1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \psi_1(k) \psi_2(k) e^{\imath k x} d k$$ (18-14)

This implies that Fourier transform of a convolution is a direct product of the Fourier transforms $\psi_1(k) \psi_2(k)$ .

Another way to think of this is that ``the net effect on the spatial function due two interfering waves is contained by product the fourier transforms.'' Practically, if the effect of an aperture (i.e., a sample of only a finite part of real space) on a wave-function is desired, then it can be obtained by multiplying the Fourier transform of the aperture and the Fourier transform of the entire wave-function.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

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Creating Lattices for Subsequent Fourier Transform

A diffraction pattern from a group of scattering centers such atoms is related to the Fourier transform of the ``atom'' positions:

1. Create ``pixel images'' of lattices by placing ones (white) and zeroes (black) in a rectangular grid.
2. This can be done by creating ``white'' matrix sets and ``black'' matrix sets and then copying them periodically into the rectangular region.
3. Recursive copying operations will create a ``perfect lattice.''

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

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Discrete Fourier Transforms

A Fourier transform is over an infinite domain. Numerical data is seldom infinite, therefore a strategy must be applied to get a Fourier transform of data.

Discrete Fourier transforms (DFT) operate by creating a lattice of copies of the original data and then returning the Fourier transform of the result. Symmetry elements within the data appear in the Discrete Fourier transform and are superimposed with the Transform of the symmetry operations due to the virtual infinite lattice of data patterns.

Because there are a finite number of pixels in the data, there are also the same finite number of sub-periodic wave-numbers that can be determined. In other words, the Discrete Fourier Transform of a $N \times M$ image will be a data set of $N \times M$ wave-numbers:

$$\begin{split} \text{Discrete FT Data} = 2 \pi ( \frac{1}{N ... ...ext{pixels}} , \ldots, \frac{M}{M \text{pixels}} ) \end{split}$$

representing the amplitudes of the indicated periodicities.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

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Fourier Transforms on Lattices with Thermal Noise

Lattices in real systems not only contain defects, but also some uncertainty in the positions of the atoms because of thermal effects such as phonons.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

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Imaging from Selected Regions of Reciprocal Space

To select and interpret different regions of Fourier space, a function will be produced that selects a particular region of the Fourier Space (i.e., as selected set of possible periodicities) and then visualize the Back-Transform of only that region.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

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Taking Discrete Fourier Transforms of Images

A image in graphics format, such as a .gif, contains intensity as a function of position. If the function is gray-scale data, then each pixel typically takes on $2^8$ discrete gray values between 0 and 255. This data can be input into MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ and then Fourier transformed.