\documentclass{article}
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\begin{document}

\begin{center}
Low Pass Filters

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Space
\end{center}

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average: \ \ \ new = $\frac{x_{k+1}+x_{k-1}}{2}$

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ binomial \ \ \ 
$\frac{ x_{k+1}+2x_{k}+x_{k-1}}{4}$\ 

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\bigskip\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ Pascal

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ 1

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ 1 \ \ \ \ \ 2 \ \ \ \ \ \ 1

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ 1 \ \ \ \ 3 \ \ \ \ \ \ 3 \ \ \ \ \ 1

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1
\ \ \ \ 4 \ \ \ \ \ 6 \ \ \ \ \ 4 \ \ \ \ \ 1

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \
\ 5 \ \ \ 10 \ \ \ \  10 \ \ \ \  5 \ \ \ \ \ 1

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ 6 \ \
\ \ 15 \ \ \ 20 \ \ \ 15 \ \ \ \ 6 \ \ \ \ \ \ 1

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\begin{center}
Fourier
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\begin{center}
Ideal
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\begin{equation*}
H(u,v)=%
\begin{cases}
1 & \sqrt{u^{2}+v^{2}}\leq R^{2} \\ 
0 & \sqrt{u^{2}+v^{2}}\geq R^{2}%
\end{cases}%
\end{equation*}%
or 
\begin{equation*}
H(u,v)=%
\begin{cases}
1 & |u|\leq u_{0}\text{ and }|v|\leq v_{0} \\ 
0 & \text{otherwise}%
\end{cases}%
\end{equation*}

\begin{center}
Butterworth
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\begin{equation*}
H(u,v)=\frac{1}{1+\left( \frac{\sqrt{u^{2}+v^{2}}}{R^{2}}\right) ^{2K}}
\end{equation*}

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\begin{center}
Gaussian
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\begin{align*}
H(u,v) &= e^{-2\pi^2\sigma^2 (u^2+v^2)} \\
h(m,n) &= \frac{1}{2\pi \sigma }e^{-\frac{m^2+n^2}{2\sigma^2}}
\end{align*}

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