| Version current |
Version 3 |
| Below are tables of \PMlinkname{Fourier transforms}{FourierTransform}; one lists some of the common properties, and the other lists some common examples. |
Below are tables of \PMlinkname{Fourier transforms}{FourierTransform}; one lists some of the common properties, and the other lists some common examples. |
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| \subsubsection*{Properties} |
\subsubsection*{Properties} |
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| \begin{center} |
\begin{center} |
| \begin{tabular}{|c|c|p{4cm}|c|} |
\begin{tabular}{|c|c|p{4cm}|c|} |
| \hline\hline |
\hline\hline |
| Original & Transformed & comment & derivation \\ |
Original & Transformed & comment & derivation \\ |
| \hline\hline |
\hline\hline |
| $af(t)+bg(t)$ & $a\F{f(t)}+b\F{g(t)}$ & linearity & \\ |
$af(t)+bg(t)$ & $a\F{f(t)}+b\F{g(t)}$ & linearity & \\ |
| \hline |
\hline |
| $f(t)*g(t)$ & $\F{f(t)}\F{g(t)}$ & convolution property & \\ |
$f(t)*g(t)$ & $\F{f(t)}\F{g(t)}$ & convolution property & \\ |
| \hline |
\hline |
| $f(t+\alpha)$ & $F(s)\exp(-i \alpha s)$ & time shift, where $F(s)=\F{f(t)}$ & \\ |
$f(t+\alpha)$ & $F(s)\exp(-i \alpha s)$ & time shift, where $F(s)=\F{f(t)}$ & \\ |
| \hline |
\hline |
| $f'(t)$ & $is \F{f(t)}$ & differentiation & \\ |
$f'(t)$ & $is \F{f(t)}$ & differentiation & \\ |
| \hline |
\hline |
| $\overline{f(t)}$ & $\overline{F(-s)}$ & conjugation, where $F(s)=\F{f(t)}$ & \\ |
$\overline{f(t)}$ & $\overline{F(-s)}$ & conjugation, where $F(s)=\F{f(t)}$ & \\ |
| \hline |
\hline |
| $f(\alpha t)$ & $\displaystyle{\frac{1}{|\alpha|}F(\frac{s}{\alpha})}$ & scaling, where $F(s)=\F{f(t)}$ with $\alpha\ne 0$ & \\ |
$f(\alpha t)$ & $\displaystyle{\frac{1}{|\alpha|}F(\frac{s}{\alpha})}$ & scaling, where $F(s)=\F{f(t)}$ with $\alpha\ne 0$ & \\ |
| \hline |
\hline |
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| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
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| \subsubsection*{Examples} |
\subsubsection*{Examples} |
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| \begin{center} |
\begin{center} |
| \begin{tabular}{|c|c|c|p{4cm}|c|} |
\begin{tabular}{|c|c|c|p{4cm}|c|} |
| \hline\hline |
\hline\hline |
| $f(t)$ & $\F{f(t)}$ & conditions & explanation & derivation \\ |
$f(t)$ & $\F{f(t)}$ & conditions & explanation & derivation \\ |
| \hline\hline |
\hline\hline |
| $\delta(t)$ & $1$ & & Dirac delta function & \\ |
$\delta(t)$ & $1$ & & Dirac delta function & \\ |
| \hline |
\hline |
| $1$ & $2\pi \delta(s)$ & & & \\ |
$1$ & $2\pi \delta(s)$ & & & \\ |
| \hline |
\hline |
| $e^{i a t}$ & $2\pi \delta(s - \alpha)$ & $a\in \mathbb{R}$ & & \\ |
$e^{i a t}$ & $2\pi \delta(s - \alpha)$ & $a\in \mathbb{R}$ & & \\ |
| \hline |
\hline |
| $\cos(at)$ & $\pi (\delta(s+a) + \delta(s-a))$ & $a\in \mathbb{R}$ & &\\ |
$\cos(at)$ & $\pi (\delta(s+a) + \delta(s-a))$ & $a\in \mathbb{R}$ & &\\ |
| \hline |
\hline |
| $\sin(at)$ & $i \pi (\delta(s+a) - \delta(s-a))$ & $a\in \mathbb{R}$ & &\\ |
$\sin(at)$ & $i \pi (\delta(s+a) - \delta(s-a))$ & $a\in \mathbb{R}$ & &\\ |
| \hline |
\hline |
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|
| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
