\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} %TODO plot this \draw[help lines, dashed] (-7,-1) grid (7,5); \draw[-latex] (-7,0) -- (7,0) node[below]{$f$}; \draw[-latex] (0,-1) -- (0,5) node[left]{$S_x(f)$}; % T = 3 \draw[very thick, teal, smooth] plot[domain=-7:7] (-7.0, 9.236424985855951e-66) -- (-6.9, 1.351508856906833e-07) -- (-6.8, 9.82042242286396e-07) -- (-6.7, 1.0419973375281967e-06) -- (-6.6, 1.6145054556795375e-07) -- (-6.5, 2.1247489470901966e-65) -- (-6.4, 1.825977253441778e-07) -- (-6.3, 1.3329185814786595e-06) -- (-6.2, 1.4210163081164671e-06) -- (-6.1, 2.2125643933133683e-07) -- (-6.0, 9.236424985855951e-66) -- (-5.9, 2.5281754659351027e-07) -- (-5.8, 1.8554696109297905e-06) -- (-5.7, 1.9891447935831516e-06) -- (-5.6, 3.1150365806319875e-07) -- (-5.5, 1.6166354311481645e-63) -- (-5.4, 3.602800545131002e-07) -- (-5.3, 2.661107026328134e-06) -- (-5.2, 2.8717884569168376e-06) -- (-5.1, 4.528293363652966e-07) -- (-5.0, 9.236424985855958e-66) -- (-4.9, 5.31411488307728e-07) -- (-4.8, 3.95549527961044e-06) -- (-4.7, 4.303030142791807e-06) -- (-4.6, 6.842013588090742e-07) -- (-4.5, 9.236424985855951e-66) -- (-4.4, 8.173433869377619e-07) -- (-4.3, 6.141748066122749e-06) -- (-4.2, 6.747900318735658e-06) -- (-4.1, 1.0841261802423652e-06) -- (-4.0, 9.236424985855951e-66) -- (-3.9, 1.3242099017039712e-06) -- (-3.8, 1.0070045517514694e-05) -- (-3.7, 1.1203634282331384e-05) -- (-3.6, 1.8239177759725472e-06) -- (-3.5, 9.236424985855951e-66) -- (-3.4, 2.2924485153493843e-06) -- (-3.3, 1.7705576029270517e-05) -- (-3.2, 2.0024694853027794e-05) -- (-3.1, 3.3171757645011927e-06) -- (-3.0, 9.236424985855951e-66) -- (-2.9, 4.331349886689919e-06) -- (-2.8, 3.4161245363599325e-05) -- (-2.7, 3.951033968062021e-05) -- (-2.6, 6.70381262737629e-06) -- (-2.5, 9.236424985855958e-66) -- (-2.4, 9.233583740861088e-06) -- (-2.3, 7.503337405958586e-05) -- (-2.2, 8.963447864818203e-05) -- (-2.1, 1.5752086215146527e-05) -- (-2.0, 9.236424985855951e-66) -- (-1.9, 2.3507197452505176e-05) -- (-1.8, 0.0002000210946331401) -- (-1.7, 0.00025140281402531684) -- (-1.6, 4.6745017688109035e-05) -- (-1.5, 9.236424985855951e-66) -- (-1.4, 7.974493646417963e-05) -- (-1.3, 0.0007351778449707119) -- (-1.2, 0.0010126067915802706) -- (-1.1, 0.00020923990705607045) -- (-1.0, 9.236424985855951e-66) -- (-0.9, 0.0004669229506489748) -- (-0.8, 0.005126321882375121) -- (-0.7, 0.00874527881308142) -- (-0.6, 0.002363797437660431) -- (-0.5, 9.236424985855951e-66) -- (-0.4, 0.011966724528155946) -- (-0.3, 0.2592273386445493) -- (-0.2, 1.312338401888031) -- (-0.1, 3.06348147920792) -- (0.0, 4) -- (0.1, 3.06348147920792) -- (0.2, 1.312338401888031) -- (0.3, 0.2592273386445493) -- (0.4, 0.011966724528155946) -- (0.5, 9.236424985855951e-66) -- (0.6, 0.002363797437660431) -- (0.7, 0.00874527881308142) -- (0.8, 0.005126321882375121) -- (0.9, 0.0004669229506489748) -- (1.0, 9.236424985855951e-66) -- (1.1, 0.00020923990705607045) -- (1.2, 0.0010126067915802706) -- (1.3, 0.0007351778449707119) -- (1.4, 7.974493646417963e-05) -- (1.5, 9.236424985855951e-66) -- (1.6, 4.6745017688109035e-05) -- (1.7, 0.00025140281402531684) -- (1.8, 0.0002000210946331401) -- (1.9, 2.3507197452505176e-05) -- (2.0, 9.236424985855951e-66) -- (2.1, 1.5752086215146527e-05) -- (2.2, 8.963447864818203e-05) -- (2.3, 7.503337405958586e-05) -- (2.4, 9.233583740861088e-06) -- (2.5, 9.236424985855958e-66) -- (2.6, 6.70381262737629e-06) -- (2.7, 3.951033968062021e-05) -- (2.8, 3.4161245363599325e-05) -- (2.9, 4.331349886689919e-06) -- (3.0, 9.236424985855951e-66) -- (3.1, 3.3171757645011927e-06) -- (3.2, 2.0024694853027794e-05) -- (3.3, 1.7705576029270517e-05) -- (3.4, 2.2924485153493843e-06) -- (3.5, 9.236424985855951e-66) -- (3.6, 1.8239177759725472e-06) -- (3.7, 1.1203634282331384e-05) -- (3.8, 1.0070045517514694e-05) -- (3.9, 1.3242099017039712e-06) -- (4.0, 9.236424985855951e-66) -- (4.1, 1.0841261802423652e-06) -- (4.2, 6.747900318735658e-06) -- (4.3, 6.141748066122749e-06) -- (4.4, 8.173433869377619e-07) -- (4.5, 9.236424985855951e-66) -- (4.6, 6.842013588090742e-07) -- (4.7, 4.303030142791807e-06) -- (4.8, 3.95549527961044e-06) -- (4.9, 5.31411488307728e-07) -- (5.0, 9.236424985855958e-66) -- (5.1, 4.528293363652966e-07) -- (5.2, 2.8717884569168376e-06) -- (5.3, 2.661107026328134e-06) -- (5.4, 3.602800545131002e-07) -- (5.5, 1.6166354311481645e-63) -- (5.6, 3.1150365806319875e-07) -- (5.7, 1.9891447935831516e-06) -- (5.8, 1.8554696109297905e-06) -- (5.9, 2.5281754659351027e-07) -- (6.0, 9.236424985855951e-66) -- (6.1, 2.2125643933133683e-07) -- (6.2, 1.4210163081164671e-06) -- (6.3, 1.3329185814786595e-06) -- (6.4, 1.825977253441778e-07) -- (6.5, 2.1247489470901966e-65) -- (6.6, 1.6145054556795375e-07) -- (6.7, 1.0419973375281967e-06) -- (6.8, 9.82042242286396e-07) -- (6.9, 1.351508856906833e-07) -- (7.0, 9.236424985855951e-66) ; \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}