334 lines
13 KiB
TeX
334 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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\draw[help lines, dashed] (-7,-1) grid (7,1);
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\draw[-latex] (-7,0) -- (7,0) node[below]{$f$};
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\draw[-latex] (0,-1) -- (0,1) node[left]{$S_x(f)$};
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% T = 1.2, A = 0.7
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\draw[very thick, teal, smooth]
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(-7.0, 1.19032961622497e-06) --
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(-6.9, 5.431980015413764e-07) --
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(-6.8, 8.800322510979997e-08) --
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(-6.7, 4.277596978633154e-10) --
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(-6.6, 7.041916257559858e-09) --
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(-6.5, 2.335906266605805e-07) --
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(-6.4, 1.058178211950344e-06) --
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(-6.3, 2.064552487744668e-06) --
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(-6.2, 2.2010067192499278e-06) --
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(-6.1, 1.2822106239978093e-06) --
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(-6.0, 3.2173909568155725e-07) --
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(-5.9, 1.102702988887616e-08) --
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(-5.8, 7.617055164926182e-10) --
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(-5.7, 1.782521662592917e-07) --
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(-5.6, 1.2519944924372998e-06) --
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(-5.5, 3.1232636115633354e-06) --
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(-5.4, 4.075939672840986e-06) --
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(-5.3, 2.9675150161786594e-06) --
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(-5.2, 1.0491415268807952e-06) --
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(-5.1, 9.482711337003691e-08) --
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(-5.0, 1.6293053675049896e-66) --
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(-4.9, 1.1128302298693327e-07) --
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(-4.8, 1.445048762984305e-06) --
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(-4.7, 4.798494174593094e-06) --
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(-4.6, 7.740543578940507e-06) --
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(-4.5, 6.969623919661404e-06) --
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(-4.4, 3.285063890545115e-06) --
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(-4.3, 5.503771776377975e-07) --
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(-4.2, 2.7701412446994334e-09) --
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(-4.1, 4.7285846864368735e-08) --
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(-4.0, 1.6288041718878957e-06) --
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(-3.9, 7.673973284118631e-06) --
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(-3.8, 1.5597454948691434e-05) --
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(-3.7, 1.735326624654691e-05) --
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(-3.6, 1.0569847172439745e-05) --
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(-3.5, 2.778668008351847e-06) --
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(-3.4, 9.998870188433106e-08) --
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(-3.3, 7.268475244613286e-09) --
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(-3.2, 1.7944622471669147e-06) --
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(-3.1, 1.3332382076743845e-05) --
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(-3.0, 3.528372109328591e-05) --
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(-2.9, 4.9001660288893075e-05) --
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(-2.8, 3.809467548087465e-05) --
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(-2.7, 1.443418925940238e-05) --
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(-2.6, 1.4038472090396448e-06) --
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(-2.5, 1.6293053675049896e-66) --
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(-2.4, 1.93360725971169e-06) --
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(-2.3, 2.741170869960495e-05) --
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(-2.2, 9.995526625731038e-05) --
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(-2.1, 0.00017820734822830307) --
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(-2.0, 0.0001786238380347597) --
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(-1.9, 9.448005177904175e-05) --
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(-1.8, 1.7924383147436297e-05) --
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(-1.7, 1.0320563008784123e-07) --
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(-1.6, 2.0388565356643826e-06) --
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(-1.5, 8.236520849447914e-05) --
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(-1.4, 0.00046213256008914604) --
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(-1.3, 0.0011387141494308545) --
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(-1.2, 0.0015684227826916776) --
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(-1.1, 0.0012125732141509873) --
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(-1.0, 0.00041697386800329994) --
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(-0.9, 2.0365569565813037e-05) --
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(-0.8, 2.1044524976969454e-06) --
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(-0.7, 0.0007836859830426006) --
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(-0.6, 0.009500566996833743) --
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(-0.5, 0.045727702536898486) --
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(-0.4, 0.13538259097964303) --
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(-0.3, 0.28907556607867346) --
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(-0.2, 0.4794324983878811) --
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(-0.1, 0.6415244821981052) --
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(0.0, 0.7055999999999999) --
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(0.1, 0.6415244821981052) --
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(0.2, 0.4794324983878811) --
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(0.3, 0.28907556607867346) --
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(0.4, 0.13538259097964303) --
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(0.5, 0.045727702536898486) --
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(0.6, 0.009500566996833743) --
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(0.7, 0.0007836859830426006) --
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(0.8, 2.1044524976969454e-06) --
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(0.9, 2.0365569565813037e-05) --
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(1.0, 0.00041697386800329994) --
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(1.1, 0.0012125732141509873) --
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(1.2, 0.0015684227826916776) --
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(1.3, 0.0011387141494308545) --
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(1.4, 0.00046213256008914604) --
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(1.5, 8.236520849447914e-05) --
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(1.6, 2.0388565356643826e-06) --
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(1.7, 1.0320563008784123e-07) --
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(1.8, 1.7924383147436297e-05) --
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(1.9, 9.448005177904175e-05) --
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(2.0, 0.0001786238380347597) --
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(2.1, 0.00017820734822830307) --
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(2.2, 9.995526625731038e-05) --
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(2.3, 2.741170869960495e-05) --
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(2.4, 1.93360725971169e-06) --
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(2.5, 1.6293053675049896e-66) --
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(2.6, 1.4038472090396448e-06) --
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(2.7, 1.443418925940238e-05) --
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(2.8, 3.809467548087465e-05) --
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(2.9, 4.9001660288893075e-05) --
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(3.0, 3.528372109328591e-05) --
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(3.1, 1.3332382076743845e-05) --
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(3.2, 1.7944622471669147e-06) --
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(3.3, 7.268475244613286e-09) --
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(3.4, 9.998870188433106e-08) --
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(3.5, 2.778668008351847e-06) --
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(3.6, 1.0569847172439745e-05) --
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(3.7, 1.735326624654691e-05) --
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(3.8, 1.5597454948691434e-05) --
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(3.9, 7.673973284118631e-06) --
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(4.0, 1.6288041718878957e-06) --
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(4.1, 4.7285846864368735e-08) --
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(4.2, 2.7701412446994334e-09) --
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(4.3, 5.503771776377975e-07) --
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(4.4, 3.285063890545115e-06) --
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(4.5, 6.969623919661404e-06) --
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(4.6, 7.740543578940507e-06) --
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(4.7, 4.798494174593094e-06) --
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(4.8, 1.445048762984305e-06) --
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(4.9, 1.1128302298693327e-07) --
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(5.0, 1.6293053675049896e-66) --
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(5.1, 9.482711337003691e-08) --
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(5.2, 1.0491415268807952e-06) --
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(5.3, 2.9675150161786594e-06) --
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(5.4, 4.075939672840986e-06) --
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(5.5, 3.1232636115633354e-06) --
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(5.6, 1.2519944924372998e-06) --
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(5.7, 1.782521662592917e-07) --
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(5.8, 7.617055164926182e-10) --
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(5.9, 1.102702988887616e-08) --
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(6.0, 3.2173909568155725e-07) --
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(6.1, 1.2822106239978093e-06) --
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(6.2, 2.2010067192499278e-06) --
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(6.3, 2.064552487744668e-06) --
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(6.4, 1.058178211950344e-06) --
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(6.5, 2.335906266605805e-07) --
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(6.6, 7.041916257559858e-09) --
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(6.7, 4.277596978633154e-10) --
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(6.8, 8.800322510979997e-08) --
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(6.9, 5.431980015413764e-07) --
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(7.0, 1.19032961622497e-06)
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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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