337 lines
13 KiB
TeX
337 lines
13 KiB
TeX
\documentclass[a4paper,french,11pt]{article}
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\title{
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Théorie du signal --- TD1
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\\ \large Décomposition en Série de Fourier
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}
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\author{}
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\date{Dernière compilation~: \today{} à \currenttime}
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\usepackage{../../cours}
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\usepackage{enumitem}
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\usepackage{xfrac}
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\usepackage{tikz}
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\begin{document}
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\maketitle
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\section{I}
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\subsection{Signal carré}
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Soit le signal carré $x(t)$, $T_0$-périodique tel que~:
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\begin{align*}
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x(t) =
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\left\{
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\begin{array}{ll}
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-1 \quad \forall\, t \in [-\frac{T_0}{2};0] \\
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1 \quad \forall\, t \in [0;\frac{T_0}{2}] \\
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\end{array}
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\right.
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\end{align*}
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\subsubsection{Tracer le signal $x(t)$}
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\begin{center}
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\begin{tikzpicture}
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\draw[help lines, dashed] (-7,-2) grid (7,2);
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\draw[-latex] (-7,0) -- (7,0) node[below]{$t$};
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\draw[-latex] (0,-2) -- (0,2) node[left]{$x(t)$};
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\foreach \i in {-6, -4, -2, 0, 2, 4, 6}{
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\draw[very thick, teal]
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(\i-1,1) -- (\i-1,-1)
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plot[domain=\i-1:\i]({\x}, {-1})
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(\i,1) -- (\i,-1)
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plot[domain=\i:\i+1]({\x}, {1})
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;
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}
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\end{tikzpicture}
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\end{center}
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\subsubsection{Calculer les coefficients de Fourier réels $a_0, a_n, b_n$ du signal $x(t)$}
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$x$ est impaire donc $a_0 = a_n = 0$.
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\begin{tabularx}{\linewidth}{XX}
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{
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\begin{equation*}
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\text{formule~:} \quad b_n = \frac{2}{T_0} \int_{(T_0)} x(t) \sin(n\omega_0 t) \dif t
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\end{equation*}
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\vspace{4cm}
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\begin{equation*}
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x(t) = \sum_{n=1}^{+\infty} \frac{2}{n\pi}(1 - (-1)^n) \sin(n\omega_0 t)
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\end{equation*}
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} &
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{
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\begin{align*}
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b_n &= \frac{4}{T_0} \int_0^{T_0/2} 1 \sin(n\omega_0 t) \dif t \\
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&= \frac{4}{T_0} \left[\frac{-\cos(n\omega_0 t)}{n\omega_0}\right]_0^{T_0/2} \\
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&= \frac{4}{T_0} \left(\frac{-\cos(n2\pi f_0 \frac{T_0}{2}) + 1}{n2\pi f_0}\right) \\
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&= 2 \left(\frac{-\cos(n\pi) + 1}{n\pi}\right) \\
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&= \frac{2}{n\pi} (-\cos(n\pi) + 1) \\
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&= \frac{2}{n\pi} (-(-1)^n + 1) \\
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b_n &= \boxed{\frac{2}{n\pi} (1 -(-1)^n)}
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\end{align*}
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} \\
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\end{tabularx}
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\subsubsection{Tracer la DSP du signal $x(t)$}
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\begin{align*}
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|c_n|^2 = |-j\frac{1}{2}b_n|^2 = \frac{1}{4}b_n^2 = \frac{1}{4}(\frac{2}{n\pi})^2(1 - (-1)^n)^2
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\end{align*}
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\begin{align*}
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S_x(f) = \sum_{-\infty}^{+\infty} |c_n|^2 \implies
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\left\{
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\begin{array}{l}
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\text{si } n \text{ est pair, } b_n = 0 \implies |c_n|^2 = 0 \\
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\text{si } n \text{ est impair, } b_n = \frac{4}{n\pi} \implies |c_n|^2 = \frac{4}{(n\pi)^2} \\
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\end{array}
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\right.
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\end{align*}
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\subsection{Signal en dent de scie}
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Soit le signal $x(t)$, $T_0$-périodique tel que~:
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\begin{align*}
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x(t) = A \times \frac{1}{T_0}t \quad \forall\, t \in [0;T_0] \\
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\end{align*}
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\subsubsection{Tracer le signal $x(t)$}
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\begin{center}
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\begin{tikzpicture}
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\draw[help lines, dashed] (-8,-2) grid (8,2);
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\draw[-latex] (-8,0) -- (8,0) node[below]{$t$};
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\draw[-latex] (0,-2) -- (0,2) node[left]{$x(t)$};
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\foreach \i in {-8, -6, -4, -2, 0, 2, 4, 6}{
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\draw[very thick, teal]
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plot[domain=\i:\i+2]({\x}, {1.7*(\x-\i)/2})
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(\i+2,0) -- (\i+2,1.7)
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;
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}
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\end{tikzpicture}
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\end{center}
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\subsection{Signal porte}
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Soit le signal $x(t)$, de largeur $T>0$ tel que~:
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\begin{align*}
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x(t) = A\Pi_r(t) =
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\left\{
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\begin{array}{ll}
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A \quad \forall\, t \in [-\frac{T}{2};\frac{T}{2}] \\
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0 \text{ sinon} \\
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\end{array}
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\right.
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\end{align*}
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\subsubsection{Calculer $X(f)$, la TF de $x(t)$. Représenter la DSE de $x(t)$}
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\begin{equation*}
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X(f) = \int_{\mathbb{R}} x(t) e^{-j2\pi ft} \dif t
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\end{equation*}
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\begin{align*}
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X(f) &= \int_{-T/2}^{T/2} A e^{-j2\pi ft} \dif t \\
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&= A \left[\frac{e^{-j2\pi ft}}{-j2\pi f}\right]_{-T/2}^{T/2} \\
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&= A \left(\frac{-e^{-j\pi fT} - e^{j\pi fT}}{-j2\pi f}\right) \\
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&= A \left(\frac{e^{j\pi fT} - e^{-j\pi fT}}{-j2\pi f}\right) \\
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&= A \textcolor{red}{T} \frac{\sin(\pi fT)}{\pi f \textcolor{red}{T}} \\
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X(f) &= \boxed{AT \text{sinc}(fT)}
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\end{align*}
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\paragraph{DSE}
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\begin{align*}
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S_x(f) &= |X(f)|^2 \\
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&= A^2T^2\text{sinc}^2(fT)
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\end{align*}
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\begin{align*}
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\text{Rappel~: }
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\left\{
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\begin{array}{l}
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\text{sinc}(0) = 1 \\
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\text{sinc}(fT) = \frac{\sin(\pi fT)}{\pi fT} \\
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\end{array}
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\right.
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\end{align*}
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\begin{center}
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\begin{tikzpicture}
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%TODO plot this
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\draw[help lines, dashed] (-7,-1) grid (7,5);
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\draw[-latex] (-7,0) -- (7,0) node[below]{$f$};
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\draw[-latex] (0,-1) -- (0,5) node[left]{$S_x(f)$};
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% T = 3
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\draw[very thick, teal, smooth]
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plot[domain=-7:7]
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(-7.0, 9.236424985855951e-66) --
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(-6.9, 1.351508856906833e-07) --
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(-6.8, 9.82042242286396e-07) --
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(-6.7, 1.0419973375281967e-06) --
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(-6.6, 1.6145054556795375e-07) --
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(-6.5, 2.1247489470901966e-65) --
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(-6.4, 1.825977253441778e-07) --
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(-6.3, 1.3329185814786595e-06) --
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(-6.2, 1.4210163081164671e-06) --
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(-6.1, 2.2125643933133683e-07) --
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(-6.0, 9.236424985855951e-66) --
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(-5.9, 2.5281754659351027e-07) --
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(-5.8, 1.8554696109297905e-06) --
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(-5.7, 1.9891447935831516e-06) --
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(-5.6, 3.1150365806319875e-07) --
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(-5.5, 1.6166354311481645e-63) --
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(-5.4, 3.602800545131002e-07) --
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(-5.3, 2.661107026328134e-06) --
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(-5.2, 2.8717884569168376e-06) --
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(-5.1, 4.528293363652966e-07) --
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(-5.0, 9.236424985855958e-66) --
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(-4.9, 5.31411488307728e-07) --
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(-4.8, 3.95549527961044e-06) --
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(-4.7, 4.303030142791807e-06) --
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(-4.6, 6.842013588090742e-07) --
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(-4.5, 9.236424985855951e-66) --
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(-4.4, 8.173433869377619e-07) --
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(-4.3, 6.141748066122749e-06) --
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(-4.2, 6.747900318735658e-06) --
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(-4.1, 1.0841261802423652e-06) --
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(-4.0, 9.236424985855951e-66) --
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(-3.9, 1.3242099017039712e-06) --
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(-3.8, 1.0070045517514694e-05) --
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(-3.7, 1.1203634282331384e-05) --
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(-3.6, 1.8239177759725472e-06) --
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(-3.5, 9.236424985855951e-66) --
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(-3.4, 2.2924485153493843e-06) --
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(-3.3, 1.7705576029270517e-05) --
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(-3.2, 2.0024694853027794e-05) --
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(-3.1, 3.3171757645011927e-06) --
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(-3.0, 9.236424985855951e-66) --
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(-2.9, 4.331349886689919e-06) --
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(-2.8, 3.4161245363599325e-05) --
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(-2.7, 3.951033968062021e-05) --
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(-2.6, 6.70381262737629e-06) --
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(-2.5, 9.236424985855958e-66) --
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(-2.4, 9.233583740861088e-06) --
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(-2.3, 7.503337405958586e-05) --
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(-2.2, 8.963447864818203e-05) --
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(-2.1, 1.5752086215146527e-05) --
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(-2.0, 9.236424985855951e-66) --
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(-1.9, 2.3507197452505176e-05) --
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(-1.8, 0.0002000210946331401) --
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(-1.7, 0.00025140281402531684) --
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(-1.6, 4.6745017688109035e-05) --
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(-1.5, 9.236424985855951e-66) --
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(-1.4, 7.974493646417963e-05) --
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(-1.3, 0.0007351778449707119) --
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(-1.2, 0.0010126067915802706) --
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(-1.1, 0.00020923990705607045) --
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(-1.0, 9.236424985855951e-66) --
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(-0.9, 0.0004669229506489748) --
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(-0.8, 0.005126321882375121) --
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(-0.7, 0.00874527881308142) --
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(-0.6, 0.002363797437660431) --
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(-0.5, 9.236424985855951e-66) --
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(-0.4, 0.011966724528155946) --
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(-0.3, 0.2592273386445493) --
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(-0.2, 1.312338401888031) --
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(-0.1, 3.06348147920792) --
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(0.0, 4) --
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(0.1, 3.06348147920792) --
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(0.2, 1.312338401888031) --
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(0.3, 0.2592273386445493) --
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(0.4, 0.011966724528155946) --
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(0.5, 9.236424985855951e-66) --
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(0.6, 0.002363797437660431) --
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(0.7, 0.00874527881308142) --
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(0.8, 0.005126321882375121) --
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(0.9, 0.0004669229506489748) --
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(1.0, 9.236424985855951e-66) --
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(1.1, 0.00020923990705607045) --
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(1.2, 0.0010126067915802706) --
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(1.3, 0.0007351778449707119) --
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(1.4, 7.974493646417963e-05) --
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(1.5, 9.236424985855951e-66) --
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(1.6, 4.6745017688109035e-05) --
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(1.7, 0.00025140281402531684) --
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(1.8, 0.0002000210946331401) --
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(1.9, 2.3507197452505176e-05) --
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(2.0, 9.236424985855951e-66) --
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(2.1, 1.5752086215146527e-05) --
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(2.2, 8.963447864818203e-05) --
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(2.3, 7.503337405958586e-05) --
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(2.4, 9.233583740861088e-06) --
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(2.5, 9.236424985855958e-66) --
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(2.6, 6.70381262737629e-06) --
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(2.7, 3.951033968062021e-05) --
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(2.8, 3.4161245363599325e-05) --
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(2.9, 4.331349886689919e-06) --
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(3.0, 9.236424985855951e-66) --
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(3.1, 3.3171757645011927e-06) --
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(3.2, 2.0024694853027794e-05) --
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(3.3, 1.7705576029270517e-05) --
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(3.4, 2.2924485153493843e-06) --
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(3.5, 9.236424985855951e-66) --
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(3.6, 1.8239177759725472e-06) --
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(3.7, 1.1203634282331384e-05) --
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(3.8, 1.0070045517514694e-05) --
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(3.9, 1.3242099017039712e-06) --
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(4.0, 9.236424985855951e-66) --
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(4.1, 1.0841261802423652e-06) --
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(4.2, 6.747900318735658e-06) --
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(4.3, 6.141748066122749e-06) --
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(4.4, 8.173433869377619e-07) --
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(4.5, 9.236424985855951e-66) --
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(4.6, 6.842013588090742e-07) --
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(4.7, 4.303030142791807e-06) --
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(4.8, 3.95549527961044e-06) --
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(4.9, 5.31411488307728e-07) --
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(5.0, 9.236424985855958e-66) --
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(5.1, 4.528293363652966e-07) --
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(5.2, 2.8717884569168376e-06) --
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(5.3, 2.661107026328134e-06) --
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(5.4, 3.602800545131002e-07) --
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(5.5, 1.6166354311481645e-63) --
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(5.6, 3.1150365806319875e-07) --
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(5.7, 1.9891447935831516e-06) --
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(5.8, 1.8554696109297905e-06) --
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(5.9, 2.5281754659351027e-07) --
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(6.0, 9.236424985855951e-66) --
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(6.1, 2.2125643933133683e-07) --
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(6.2, 1.4210163081164671e-06) --
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(6.3, 1.3329185814786595e-06) --
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(6.4, 1.825977253441778e-07) --
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(6.5, 2.1247489470901966e-65) --
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(6.6, 1.6145054556795375e-07) --
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(6.7, 1.0419973375281967e-06) --
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(6.8, 9.82042242286396e-07) --
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(6.9, 1.351508856906833e-07) --
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(7.0, 9.236424985855951e-66)
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;
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\end{tikzpicture}
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\end{center}
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\subsubsection{Que vaut $\int_{\mathbb{R}} S_x(f) \dif f$~?}
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Calculer l'intégrale dans le domaine fréquentiel serait compliqué.
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Mais d'après le théorême de Parseval~:
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\begin{align*}
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E = \int_{\mathbb{R}} S_x(f) \dif f
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&= \int_{\mathbb{R}} |x(t)|^2 \dif t \\
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&= \int_{-\pi/2}^{\pi/2} A^2 \dif t \\
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&= A^2 [t]_{-T/2}^{T/2} \\
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&= A^2 T
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\end{align*}
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\section{II}
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\end{document}
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