\documentclass[a4paper,french,11pt]{article} \title{ Théorie du signal --- TD1 \\ \large Décomposition en Série de Fourier } \author{} \date{Dernière compilation~: \today{} à \currenttime} \usepackage{../../cours} \usepackage{enumitem} \usepackage{xfrac} \usepackage{tikz} \begin{document} \maketitle \section{I} \subsection{Signal carré} Soit le signal carré $x(t)$, $T_0$-périodique tel que~: \begin{align*} x(t) = \left\{ \begin{array}{ll} -1 \quad \forall\, t \in [-\frac{T_0}{2};0] \\ 1 \quad \forall\, t \in [0;\frac{T_0}{2}] \\ \end{array} \right. \end{align*} \subsubsection{Tracer le signal $x(t)$} \begin{center} \begin{tikzpicture} \draw[help lines, dashed] (-7,-2) grid (7,2); \draw[-latex] (-7,0) -- (7,0) node[below]{$t$}; \draw[-latex] (0,-2) -- (0,2) node[left]{$x(t)$}; \foreach \i in {-6, -4, -2, 0, 2, 4, 6}{ \draw[very thick, teal] (\i-1,1) -- (\i-1,-1) plot[domain=\i-1:\i]({\x}, {-1}) (\i,1) -- (\i,-1) plot[domain=\i:\i+1]({\x}, {1}) ; } \end{tikzpicture} \end{center} \subsubsection{Calculer les coefficients de Fourier réels $a_0, a_n, b_n$ du signal $x(t)$} $x$ est impaire donc $a_0 = a_n = 0$. \begin{tabularx}{\linewidth}{XX} { \begin{equation*} \text{formule~:} \quad b_n = \frac{2}{T_0} \int_{(T_0)} x(t) \sin(n\omega_0 t) \dif t \end{equation*} \vspace{4cm} \begin{equation*} x(t) = \sum_{n=1}^{+\infty} \frac{2}{n\pi}(1 - (-1)^n) \sin(n\omega_0 t) \end{equation*} } & { \begin{align*} b_n &= \frac{4}{T_0} \int_0^{T_0/2} 1 \sin(n\omega_0 t) \dif t \\ &= \frac{4}{T_0} \left[\frac{-\cos(n\omega_0 t)}{n\omega_0}\right]_0^{T_0/2} \\ &= \frac{4}{T_0} \left(\frac{-\cos(n2\pi f_0 \frac{T_0}{2}) + 1}{n2\pi f_0}\right) \\ &= 2 \left(\frac{-\cos(n\pi) + 1}{n\pi}\right) \\ &= \frac{2}{n\pi} (-\cos(n\pi) + 1) \\ &= \frac{2}{n\pi} (-(-1)^n + 1) \\ b_n &= \boxed{\frac{2}{n\pi} (1 -(-1)^n)} \end{align*} } \\ \end{tabularx} \subsubsection{Tracer la DSP du signal $x(t)$} \begin{align*} |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 \end{align*} \begin{align*} S_x(f) = \sum_{-\infty}^{+\infty} |c_n|^2 \implies \left\{ \begin{array}{l} \text{si } n \text{ est pair, } b_n = 0 \implies |c_n|^2 = 0 \\ \text{si } n \text{ est impair, } b_n = \frac{4}{n\pi} \implies |c_n|^2 = \frac{4}{(n\pi)^2} \\ \end{array} \right. \end{align*} \subsection{Signal en dent de scie} Soit le signal $x(t)$, $T_0$-périodique tel que~: \begin{align*} x(t) = A \times \frac{1}{T_0}t \quad \forall\, t \in [0;T_0] \\ \end{align*} \subsubsection{Tracer le signal $x(t)$} \begin{center} \begin{tikzpicture} \draw[help lines, dashed] (-8,-2) grid (8,2); \draw[-latex] (-8,0) -- (8,0) node[below]{$t$}; \draw[-latex] (0,-2) -- (0,2) node[left]{$x(t)$}; \foreach \i in {-8, -6, -4, -2, 0, 2, 4, 6}{ \draw[very thick, teal] plot[domain=\i:\i+2]({\x}, {1.7*(\x-\i)/2}) (\i+2,0) -- (\i+2,1.7) ; } \end{tikzpicture} \end{center} \subsection{Signal porte} Soit le signal $x(t)$, de largeur $T>0$ tel que~: \begin{align*} x(t) = A\Pi_r(t) = \left\{ \begin{array}{ll} A \quad \forall\, t \in [-\frac{T}{2};\frac{T}{2}] \\ 0 \text{ sinon} \\ \end{array} \right. \end{align*} \subsubsection{Calculer $X(f)$, la TF de $x(t)$. Représenter la DSE de $x(t)$} \begin{equation*} X(f) = \int_{\mathbb{R}} x(t) e^{-j2\pi ft} \dif t \end{equation*} \begin{align*} X(f) &= \int_{-T/2}^{T/2} A e^{-j2\pi ft} \dif t \\ &= A \left[\frac{e^{-j2\pi ft}}{-j2\pi f}\right]_{-T/2}^{T/2} \\ &= A \left(\frac{-e^{-j\pi fT} - e^{j\pi fT}}{-j2\pi f}\right) \\ &= A \left(\frac{e^{j\pi fT} - e^{-j\pi fT}}{-j2\pi f}\right) \\ &= A \textcolor{red}{T} \frac{\sin(\pi fT)}{\pi f \textcolor{red}{T}} \\ X(f) &= \boxed{AT \text{sinc}(fT)} \end{align*} \paragraph{DSE} \begin{align*} S_x(f) &= |X(f)|^2 \\ &= A^2T^2\text{sinc}^2(fT) \end{align*} \begin{align*} \text{Rappel~: } \left\{ \begin{array}{l} \text{sinc}(0) = 1 \\ \text{sinc}(fT) = \frac{\sin(\pi fT)}{\pi fT} \\ \end{array} \right. \end{align*} \begin{center} \begin{tikzpicture} \draw[help lines, dashed] (-7,-1) grid (7,1); \draw[-latex] (-7,0) -- (7,0) node[below]{$f$}; \draw[-latex] (0,-1) -- (0,1) node[left]{$S_x(f)$}; % T = 1.2, A = 0.7 \draw[very thick, teal, smooth] (-7.0, 1.19032961622497e-06) -- (-6.9, 5.431980015413764e-07) -- (-6.8, 8.800322510979997e-08) -- (-6.7, 4.277596978633154e-10) -- (-6.6, 7.041916257559858e-09) -- (-6.5, 2.335906266605805e-07) -- (-6.4, 1.058178211950344e-06) -- (-6.3, 2.064552487744668e-06) -- (-6.2, 2.2010067192499278e-06) -- (-6.1, 1.2822106239978093e-06) -- (-6.0, 3.2173909568155725e-07) -- (-5.9, 1.102702988887616e-08) -- (-5.8, 7.617055164926182e-10) -- (-5.7, 1.782521662592917e-07) -- (-5.6, 1.2519944924372998e-06) -- (-5.5, 3.1232636115633354e-06) -- (-5.4, 4.075939672840986e-06) -- (-5.3, 2.9675150161786594e-06) -- (-5.2, 1.0491415268807952e-06) -- (-5.1, 9.482711337003691e-08) -- (-5.0, 1.6293053675049896e-66) -- (-4.9, 1.1128302298693327e-07) -- (-4.8, 1.445048762984305e-06) -- (-4.7, 4.798494174593094e-06) -- (-4.6, 7.740543578940507e-06) -- (-4.5, 6.969623919661404e-06) -- (-4.4, 3.285063890545115e-06) -- (-4.3, 5.503771776377975e-07) -- (-4.2, 2.7701412446994334e-09) -- (-4.1, 4.7285846864368735e-08) -- (-4.0, 1.6288041718878957e-06) -- (-3.9, 7.673973284118631e-06) -- (-3.8, 1.5597454948691434e-05) -- (-3.7, 1.735326624654691e-05) -- (-3.6, 1.0569847172439745e-05) -- (-3.5, 2.778668008351847e-06) -- (-3.4, 9.998870188433106e-08) -- (-3.3, 7.268475244613286e-09) -- (-3.2, 1.7944622471669147e-06) -- (-3.1, 1.3332382076743845e-05) -- (-3.0, 3.528372109328591e-05) -- (-2.9, 4.9001660288893075e-05) -- (-2.8, 3.809467548087465e-05) -- (-2.7, 1.443418925940238e-05) -- (-2.6, 1.4038472090396448e-06) -- (-2.5, 1.6293053675049896e-66) -- (-2.4, 1.93360725971169e-06) -- (-2.3, 2.741170869960495e-05) -- (-2.2, 9.995526625731038e-05) -- (-2.1, 0.00017820734822830307) -- (-2.0, 0.0001786238380347597) -- (-1.9, 9.448005177904175e-05) -- (-1.8, 1.7924383147436297e-05) -- (-1.7, 1.0320563008784123e-07) -- (-1.6, 2.0388565356643826e-06) -- (-1.5, 8.236520849447914e-05) -- (-1.4, 0.00046213256008914604) -- (-1.3, 0.0011387141494308545) -- (-1.2, 0.0015684227826916776) -- (-1.1, 0.0012125732141509873) -- (-1.0, 0.00041697386800329994) -- (-0.9, 2.0365569565813037e-05) -- (-0.8, 2.1044524976969454e-06) -- (-0.7, 0.0007836859830426006) -- (-0.6, 0.009500566996833743) -- (-0.5, 0.045727702536898486) -- (-0.4, 0.13538259097964303) -- (-0.3, 0.28907556607867346) -- (-0.2, 0.4794324983878811) -- (-0.1, 0.6415244821981052) -- (0.0, 0.7055999999999999) -- (0.1, 0.6415244821981052) -- (0.2, 0.4794324983878811) -- (0.3, 0.28907556607867346) -- (0.4, 0.13538259097964303) -- (0.5, 0.045727702536898486) -- (0.6, 0.009500566996833743) -- (0.7, 0.0007836859830426006) -- (0.8, 2.1044524976969454e-06) -- (0.9, 2.0365569565813037e-05) -- (1.0, 0.00041697386800329994) -- (1.1, 0.0012125732141509873) -- (1.2, 0.0015684227826916776) -- (1.3, 0.0011387141494308545) -- (1.4, 0.00046213256008914604) -- (1.5, 8.236520849447914e-05) -- (1.6, 2.0388565356643826e-06) -- (1.7, 1.0320563008784123e-07) -- (1.8, 1.7924383147436297e-05) -- (1.9, 9.448005177904175e-05) -- (2.0, 0.0001786238380347597) -- (2.1, 0.00017820734822830307) -- (2.2, 9.995526625731038e-05) -- (2.3, 2.741170869960495e-05) -- (2.4, 1.93360725971169e-06) -- (2.5, 1.6293053675049896e-66) -- (2.6, 1.4038472090396448e-06) -- (2.7, 1.443418925940238e-05) -- (2.8, 3.809467548087465e-05) -- (2.9, 4.9001660288893075e-05) -- (3.0, 3.528372109328591e-05) -- (3.1, 1.3332382076743845e-05) -- (3.2, 1.7944622471669147e-06) -- (3.3, 7.268475244613286e-09) -- (3.4, 9.998870188433106e-08) -- (3.5, 2.778668008351847e-06) -- (3.6, 1.0569847172439745e-05) -- (3.7, 1.735326624654691e-05) -- (3.8, 1.5597454948691434e-05) -- (3.9, 7.673973284118631e-06) -- (4.0, 1.6288041718878957e-06) -- (4.1, 4.7285846864368735e-08) -- (4.2, 2.7701412446994334e-09) -- (4.3, 5.503771776377975e-07) -- (4.4, 3.285063890545115e-06) -- (4.5, 6.969623919661404e-06) -- (4.6, 7.740543578940507e-06) -- (4.7, 4.798494174593094e-06) -- (4.8, 1.445048762984305e-06) -- (4.9, 1.1128302298693327e-07) -- (5.0, 1.6293053675049896e-66) -- (5.1, 9.482711337003691e-08) -- (5.2, 1.0491415268807952e-06) -- (5.3, 2.9675150161786594e-06) -- (5.4, 4.075939672840986e-06) -- (5.5, 3.1232636115633354e-06) -- (5.6, 1.2519944924372998e-06) -- (5.7, 1.782521662592917e-07) -- (5.8, 7.617055164926182e-10) -- (5.9, 1.102702988887616e-08) -- (6.0, 3.2173909568155725e-07) -- (6.1, 1.2822106239978093e-06) -- (6.2, 2.2010067192499278e-06) -- (6.3, 2.064552487744668e-06) -- (6.4, 1.058178211950344e-06) -- (6.5, 2.335906266605805e-07) -- (6.6, 7.041916257559858e-09) -- (6.7, 4.277596978633154e-10) -- (6.8, 8.800322510979997e-08) -- (6.9, 5.431980015413764e-07) -- (7.0, 1.19032961622497e-06) ; \end{tikzpicture} \end{center} \subsubsection{Que vaut $\int_{\mathbb{R}} S_x(f) \dif f$~?} Calculer l'intégrale dans le domaine fréquentiel serait compliqué. Mais d'après le théorême de Parseval~: \begin{align*} E = \int_{\mathbb{R}} S_x(f) \dif f &= \int_{\mathbb{R}} |x(t)|^2 \dif t \\ &= \int_{-\pi/2}^{\pi/2} A^2 \dif t \\ &= A^2 [t]_{-T/2}^{T/2} \\ &= A^2 T \end{align*} \section{II} \end{document}