efrei/theorie-signal/exercices/td1.tex

\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}