1.5. Formulaire et Tables des principales transformées de Fourier#
\[\begin{split}\begin{aligned}
ax(t)+by(t) & \rightleftharpoons aX(f)+bY(f) \\
x(t-t_{0}) & \rightleftharpoons X(f)e^{-i 2 \pi f t_{0}} \\
x(t)e^{+i 2 \pi f_{0} t} & \rightleftharpoons X(f-f_{0}) \\
x^{\ast }(t)& \rightleftharpoons X^{\ast}(-f) \\
x(t) y(t)& \rightleftharpoons X(f)\ast Y(f) \\
x(t)\ast y(t) & \rightleftharpoons X(f) Y(f) \\
x(at+b) & \rightleftharpoons \frac{1}{\left|a\right|}X\left(\frac{f}{a}\right) e^{i2\pi \frac{b}{a}f} \\
\frac{dx^{(n)}(t)}{dt^{n}} & \rightleftharpoons \left( i2\pi f\right) ^{n}X(f) \\
\left( -i2\pi t\right)^{n}x(t) & \rightleftharpoons \frac{dX^{(n)}(f)}{df^{n}}
\end{aligned}\end{split}\]
\[\int_{\mathbb{R}}x(t)y^{\ast }(t)dt=\int_{\mathbb{R}}X(f)Y^{\ast }(f)df\]
\[\int_{\mathbb{R}}\left| x(t)\right| ^{2}dt=\int_{\mathbb{R}}\left| X(f)\right| ^{2}df\]
\[\underset{n\in \mathbb{Z}}{\sum }c_{n}e^{+i2\pi nf_{0}t}\rightleftharpoons \underset{n\in \mathbb{Z}}{\sum }c_{n}\delta \left( f- nf_{0}\right)\]
\[\begin{split}\begin{aligned}
\hline\\
1 & \rightleftharpoons \delta \left( f \right)\\
\delta \left( t\right) & \rightleftharpoons 1 \\
e^{+i2\pi f_{0}t}& \rightleftharpoons \delta \left( f-f_{0}\right)\\
\delta \left( t-t_{0}\right) & \rightleftharpoons e^{-i2\pi ft_{0}} \\
\amalg \hspace{-0.3cm}\amalg _{T}\left( t\right) & \rightleftharpoons \frac{1}{T}\amalg \hspace{-0.3cm}\amalg _{1/T}\left(f\right) \\
\cos \left( 2 \pi f_{0}t \right) & \rightleftharpoons \frac{1}{2}\left(\delta \left( f-f_{0}\right) +\delta \left( f+f_{0}\right) \right) \\
\sin \left( 2\pi f_{0}t\right) & \rightleftharpoons \frac{1}{2i}\left( \delta \left( f-f_{0}\right)-\delta \left( f+f_{0}\right) \right)\\
e^{-a\left| t\right| } & \rightleftharpoons \frac{2a}{a^{2}+4\pi ^{2}f^{2}} \\
e^{-\pi t^{2}} & \rightleftharpoons e^{-\pi f^{2}} \\
\Pi _{T} \left( t\right) & \rightleftharpoons T \mathrm{sinc}\left( \pi Tf\right) \\
\Lambda _{T}\left( t\right) & \rightleftharpoons T \mathrm{sinc}^{2}\left( \pi Tf\right) \\
B \mathrm{ sinc}\left( \pi Bt\right) & \rightleftharpoons \Pi _{B}\left( f\right) \\
B \mathrm{ sinc}^{2}\left( \pi Bt\right) & \rightleftharpoons \Lambda _{B}\left( f\right) \\
\hline
\end{aligned}\end{split}\]
\[\amalg \hspace{-0.3cm}\amalg _{T}\left( t\right) = \underset{k\in \mathbb{Z}}{\sum } \delta \left( t-kT\right)\]
\[\mathrm{sinc}\left( \pi Tf\right)=\frac{\sin \left( \pi Tf\right) }{\pi T f}\]
Warning
\(\Pi _{T}\left( t\right)\) note une fenêtre rectangulaire de support égal à \(T\).
\(\Lambda _{T}\left( t\right)\) note une fenêtre triangulaire de support égal à \(2T\) (de demi-base égale à \(T\)).
\[\Pi _{T}\left( t\right) \ast \Pi _{T}\left( t\right) =T~\Lambda _{T}\left(
t\right)\]