Fourier Transform

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 2019/04/25   Share

This is a only a summary of knowledge.

Fourier transform

There are many conventions as in where to put the normalisation $2\pi$ factor.

Forward Fourier transform (Fourier analysis)

$\tilde{f}(k)=\int_{-\infty}^\infty f(x)e^{-ikx}\,dx$

Inverse Fourier transform (Fourier synthesis)

$f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty \tilde{f}(k) e^{ikx}\,dx$

Another common convention is to put $\frac{1}{\sqrt{2\pi}}$ before both integrals (balanced convention).

Properties

Linearity

$\begin{align} h(x)=\alpha f(x)+\beta g(x) && \iff && \tilde{h}(k)=\alpha \tilde{f}(k)+\beta \tilde{g}(k) \end{align}$

Rescaling (real $\alpha$)

$\begin{align} g(x)=f(\alpha x) && \iff && \tilde{g}(k)=\frac{1}{|\alpha|} \tilde{f}(\frac{k}{\alpha}) \end{align}$

Shift / exponential (real $\alpha$)

$\begin{align} g(x)= f(x-\alpha) && \iff && \tilde{g}(k)=e^{-ik\alpha} \tilde{f}(k)\\ g(x)= e^{ik\alpha}f(x) && \iff && \tilde{g}(k)= \tilde{f}(k-\alpha) \end{align}$

Differentiation / multiplication

$\begin{align} g(x)= f'(x) && \iff && \tilde{g}(k)=ik \tilde{f}(k)\\ g(x)= xf(x) && \iff && \tilde{g}(k)= i\tilde{f}'(k) \end{align}$

Duality

$\begin{align} g(x)= \tilde{f}(x) && \iff && \tilde{g}(k)=2\pi f(-k)\\ \end{align}$

Complex conjugation and parity inversion

$\begin{align} g(x)= [f(x)]^* && \iff && \tilde{g}(k)= [\tilde{f}(-k)]^*\\ \end{align}$

Symmetry - even & odd

$\begin{align} f(-x)= \pm f(x) && \iff && \tilde{f}(-k)=\pm \tilde{f}(k)\\ \end{align}$

FT of special functions

Top-hat function

FT is $\mathrm{sinc}$ .

e.g. $f(x)$ is a top hat of strength 1, $(-1,1)$ , then $\tilde{f}(k)=\frac{2\sin k}{k}$ .

Gaussian

FT is another Gaussian, with inversely proportional width, $\sigma_x \sigma_k=1$ .

Done by completing the square of the exponent. Also need Jordan’s lemma.

Delta function

Gives a definite position, hence should give no information about the momentum - a constant. Then we have the identity

$\int_{-\infty}^\infty e^{\pm ikx} \, dx = 2\pi \delta(k)$

Convolution theorem & such

What is convolution

Definition

The convolution $h$ of two function $f$ and $g$

$\begin{align} h=f*g && \iff && h(x)=\int_{-\infty}^\infty f(y)g(x-y) \, dx \end{align}$

This is a symmetric operation, i.e. $f*g=g*f$ .

Concept

Fix one function in space, reverse the other function in space. Then convolution is a function of the position of the reversed function, with the value of the integral of the product of the two functions.

Function / plot intuition

Copy and (continuously) pasting $f$ at $x$ with the weight $g(x)$ .

The simplist illustration of this idea is if $g$ is a delta function, or a sum of several delta functions. This is the same as “lattice & motif” idea.

Probability

If $f(x)$ and $g(y)$ are two (independent) probability distribution functions, then $h(z)$ is the PDF of $z=x+y$ .

Convolution theorem

$\begin{align} h=f*g && \iff && \tilde{h}(k)=\tilde{g}(k)\tilde{f}(k) \end{align}$

So, it is easiest to manipulate a convolution in the frequency space, as a product.

Deconvolution is possible as a division in the frequency space.

Correlation

The correlation of two functions is

$\begin{align} h=f\otimes g && \iff && h(x)=\int_{-\infty}^\infty [f(y)]^*g(x+y) \, dx \end{align}$

If two signals are in phase, the correlation is positive;
Vice versa
If two signals are completely unrelated (e.g. noises), then correlation is 0.

Parseval’s theorem

$\int_{-\infty}^\infty [f(x)]^*g(x) \, dx=\frac{1}{2\pi}\int_{-\infty}^\infty [\tilde{f}(k)]^*\tilde{g}(k) \, dk$

or more commonly, when $f=g$ ,

$\int_{-\infty}^\infty |f(x)|^2 \, dx=\frac{1}{2\pi}\int_{-\infty}^\infty |\tilde{f}(k)|^2 \, dk$

Derive from: either the Fourier transform of a correlation; or change the order of integration and use the delta function identity.

Fourier transform preserves the inner product of functions, hence it is a unitary transformation.

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Author：2LalA猪

Permalink：https://butteraddict.fun/2019/04/Fourier-Transform/

Published on：25th April 2019, 4:00 pm

Updated on：9th June 2020, 3:23 pm

License：CC BY 4.0

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