68 lines
2.3 KiB
TeX
68 lines
2.3 KiB
TeX
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\documentclass[a4paper,french,11pt]{article}
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\title{Théorie du signal --- Exercices --- Décomposition en Série de Fourier}
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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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\begin{document}
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\maketitle
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\section{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}{l}
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- 1 \,\forall\, t \in [-\frac{T_0}{2};0] \\
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1 \,\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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\subsection{Tracer le signal $x(t)$}
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\subsection{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{align*}
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b_n &= \frac{2}{T_0} \int_{(T_0)} x(t) \sin(n\omega_0 t) \dif t \\
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&= \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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x(t) = \sum_{n=1}^{+\infty} \frac{2}{n\pi}(1 - (-1)^n) \sin(2n\pi f_0 t)
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\end{align*}
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\subsection{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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\section{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 \,\forall\, t \in [0;T_0] \\
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\end{align*}
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\end{document}
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