table of Fourier transforms
TableOfFourierTransforms
2008-06-09 17:23:43
2008-07-03 19:04:44
Feature
Fourier transform
Fourier-Stieltjes generalization of FT
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Below are tables of \PMlinkname{Fourier transforms}{FourierTransform}; one lists some of the common properties, and the other lists some common examples.
\subsubsection*{Properties}
\begin{center}
\begin{tabular}{|c|c|p{4cm}|c|}
\hline\hline
Original & Transformed & comment & derivation \\
\hline\hline
$af(t)+bg(t)$ & $a\F{f(t)}+b\F{g(t)}$ & linearity & \\
\hline
$f(t)*g(t)$ & $\F{f(t)}\F{g(t)}$ & convolution property & \\
\hline
$f(t+\alpha)$ & $F(s)\exp(-i \alpha s)$ & time shift, where $F(s)=\F{f(t)}$ & \\
\hline
$f'(t)$ & $is \F{f(t)}$ & differentiation & \\
\hline
$\overline{f(t)}$ & $\overline{F(-s)}$ & conjugation, where $F(s)=\F{f(t)}$ & \\
\hline
$f(\alpha t)$ & $\displaystyle{\frac{1}{|\alpha|}F(\frac{s}{\alpha})}$ & scaling, where $F(s)=\F{f(t)}$ with $\alpha\ne 0$ & \\
\hline
\end{tabular}
\end{center}
\subsubsection*{Examples}
\begin{center}
\begin{tabular}{|c|c|c|p{4cm}|c|}
\hline\hline
$f(t)$ & $\F{f(t)}$ & conditions & explanation & derivation \\
\hline\hline
$\delta(t)$ & $1$ & & Dirac delta function & \\
\hline
$1$ & $2\pi \delta(s)$ & & & \\
\hline
$e^{i a t}$ & $2\pi \delta(s - \alpha)$ & $a\in \mathbb{R}$ & & \\
\hline
$\cos(at)$ & $\pi (\delta(s+a) + \delta(s-a))$ & $a\in \mathbb{R}$ & &\\
\hline
$\sin(at)$ & $i \pi (\delta(s+a) - \delta(s-a))$ & $a\in \mathbb{R}$ & &\\
\hline
\end{tabular}
\end{center}