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td5.tex
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\documentclass[11pt]{article}
\usepackage[breakable]{tcolorbox}
\usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour)
\usepackage{iftex}
\ifPDFTeX
\usepackage[T1]{fontenc}
\usepackage{mathpazo}
\else
\usepackage{fontspec}
\fi
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
\usepackage{graphicx}
% Maintain compatibility with old templates. Remove in nbconvert 6.0
\let\Oldincludegraphics\includegraphics
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionFormat{nocaption}{}
\captionsetup{format=nocaption,aboveskip=0pt,belowskip=0pt}
\usepackage[Export]{adjustbox} % Used to constrain images to a maximum size
\adjustboxset{max size={0.9\linewidth}{0.9\paperheight}}
\usepackage{float}
\floatplacement{figure}{H} % forces figures to be placed at the correct location
\usepackage{xcolor} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
\usepackage{textcomp} % defines textquotesingle
% Hack from http://tex.stackexchange.com/a/47451/13684:
\AtBeginDocument{%
\def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
}
\usepackage{upquote} % Upright quotes for verbatim code
\usepackage{eurosym} % defines \euro
\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
\usepackage{fancyvrb} % verbatim replacement that allows latex
\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
\makeatletter % fix for grffile with XeLaTeX
\def\Gread@@xetex#1{%
\IfFileExists{"\Gin@base".bb}%
{\Gread@eps{\Gin@base.bb}}%
{\Gread@@xetex@aux#1}%
}
\makeatother
% The hyperref package gives us a pdf with properly built
% internal navigation ('pdf bookmarks' for the table of contents,
% internal cross-reference links, web links for URLs, etc.)
\usepackage{hyperref}
% The default LaTeX title has an obnoxious amount of whitespace. By default,
% titling removes some of it. It also provides customization options.
\usepackage{titling}
\usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{booktabs} % table support for pandoc > 1.12.2
\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
% normalem makes italics be italics, not underlines
\usepackage{mathrsfs}
% Colors for the hyperref package
\definecolor{urlcolor}{rgb}{0,.145,.698}
\definecolor{linkcolor}{rgb}{.71,0.21,0.01}
\definecolor{citecolor}{rgb}{.12,.54,.11}
% ANSI colors
\definecolor{ansi-black}{HTML}{3E424D}
\definecolor{ansi-black-intense}{HTML}{282C36}
\definecolor{ansi-red}{HTML}{E75C58}
\definecolor{ansi-red-intense}{HTML}{B22B31}
\definecolor{ansi-green}{HTML}{00A250}
\definecolor{ansi-green-intense}{HTML}{007427}
\definecolor{ansi-yellow}{HTML}{DDB62B}
\definecolor{ansi-yellow-intense}{HTML}{B27D12}
\definecolor{ansi-blue}{HTML}{208FFB}
\definecolor{ansi-blue-intense}{HTML}{0065CA}
\definecolor{ansi-magenta}{HTML}{D160C4}
\definecolor{ansi-magenta-intense}{HTML}{A03196}
\definecolor{ansi-cyan}{HTML}{60C6C8}
\definecolor{ansi-cyan-intense}{HTML}{258F8F}
\definecolor{ansi-white}{HTML}{C5C1B4}
\definecolor{ansi-white-intense}{HTML}{A1A6B2}
\definecolor{ansi-default-inverse-fg}{HTML}{FFFFFF}
\definecolor{ansi-default-inverse-bg}{HTML}{000000}
% commands and environments needed by pandoc snippets
% extracted from the output of `pandoc -s`
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
% Add ',fontsize=\small' for more characters per line
\newenvironment{Shaded}{}{}
\newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
\newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
\newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
\newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
\newcommand{\RegionMarkerTok}[1]{{#1}}
\newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\NormalTok}[1]{{#1}}
% Additional commands for more recent versions of Pandoc
\newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}}
\newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}}
\newcommand{\ImportTok}[1]{{#1}}
\newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}}
\newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}}
\newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}}
\newcommand{\BuiltInTok}[1]{{#1}}
\newcommand{\ExtensionTok}[1]{{#1}}
\newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}}
\newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}}
\newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
% Define a nice break command that doesn't care if a line doesn't already
% exist.
\def\br{\hspace*{\fill} \\* }
% Math Jax compatibility definitions
\def\gt{>}
\def\lt{<}
\let\Oldtex\TeX
\let\Oldlatex\LaTeX
\renewcommand{\TeX}{\textrm{\Oldtex}}
\renewcommand{\LaTeX}{\textrm{\Oldlatex}}
% Document parameters
% Document title
\title{td5}
% Pygments definitions
\makeatletter
\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
\let\PY@ul=\relax \let\PY@tc=\relax%
\let\PY@bc=\relax \let\PY@ff=\relax}
\def\PY@tok#1{\csname PY@tok@#1\endcsname}
\def\PY@toks#1+{\ifx\relax#1\empty\else%
\PY@tok{#1}\expandafter\PY@toks\fi}
\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
\PY@it{\PY@bf{\PY@ff{#1}}}}}}}
\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
\expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
\expandafter\def\csname PY@tok@c\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
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\begin{document}
\maketitle
\hypertarget{pbs---td5}{%
\section{PBS - TD5}\label{pbs---td5}}
\hypertarget{ruxe9ception-dun-signal-bpsk-canal-gaussien-et-probabilituxe9-derreur}{%
\subsection{Réception d'un signal BPSK: canal gaussien et probabilité
d'erreur}\label{ruxe9ception-dun-signal-bpsk-canal-gaussien-et-probabilituxe9-derreur}}
\hypertarget{objectifs}{%
\subsection{Objectifs}\label{objectifs}}
\begin{itemize}
\tightlist
\item
Maitriser la relation fondamentale \(p(t) = \frac{dF}{dt}(t)\).
\item
Connaître et savoir manipuler la loi normale.
\item
Appliquer les probabilités à un problème de communication basique.
\item
Comprendre le lien entre taux d'erreur binaire et SNR.
\end{itemize}
\hypertarget{ajout-dune-constante}{%
\section{1. Ajout d'une constante}\label{ajout-dune-constante}}
Soit \(X\) une variable aléatoire \emph{quelconque}. On définit la
variable aléatoire \(Y\) de la façon suivante: \[Y = X+2.\]
\hypertarget{b-supposons-que-xsimmathcaln0-sigma_x20.25.}{%
\subsubsection{\texorpdfstring{(b) Supposons que
\(X\sim\mathcal{N}(0, \sigma_{X}^2=0.25)\).}{(b) Supposons que X\textbackslash sim\textbackslash mathcal\{N\}(0, \textbackslash sigma\_\{X\}\^{}2=0.25).}}\label{b-supposons-que-xsimmathcaln0-sigma_x20.25.}}
\hypertarget{i-quelle-est-lexpression-de-p_y}{%
\paragraph{\texorpdfstring{(i) Quelle est l'expression de \(p_Y\)
?}{(i) Quelle est l'expression de p\_Y ?}}\label{i-quelle-est-lexpression-de-p_y}}
\hypertarget{ii-donner-lallure-des-deux-densituxe9s-identifier-leur-maximum-et-les-valeurs-uxe0-123sigma-de-la-moyenne}{%
\subsection{\texorpdfstring{\#\#\#\# (ii) Donner l'allure des deux
densités, identifier leur maximum et les valeurs à \(\{1,2,3\}\sigma\)
de la
moyenne}{\#\#\#\# (ii) Donner l'allure des deux densités, identifier leur maximum et les valeurs à \textbackslash\{1,2,3\textbackslash\}\textbackslash sigma de la moyenne}}\label{ii-donner-lallure-des-deux-densituxe9s-identifier-leur-maximum-et-les-valeurs-uxe0-123sigma-de-la-moyenne}}
\hypertarget{a-densituxe9-de-probabilituxe9}{%
\subsubsection{(a) Densité de
probabilité}\label{a-densituxe9-de-probabilituxe9}}
Exprimer la densité de probabilité \(p_Y\) de \(Y\) en fonction de
\(p_X\).
\textbf{Rappel:} une densité est reliée à sa fonction de répartition par
\[p(t) = \frac{dF}{dt}(t)\]
\hypertarget{a-densituxe9-de-probabilituxe9-solution}{%
\subsubsection{(a) Densité de probabilité
(solution)}\label{a-densituxe9-de-probabilituxe9-solution}}
\begin{align}
p_Y(t) = \frac{dF_Y(t)}{dt}
&= \frac{d\mathbb{P}(Y < t)}{dt}\\
&= \frac{d\mathbb{P}(X+2 < t)}{dt}\\
&= \frac{d\mathbb{P}(X < t-2)}{dt}\\
&= \frac{dF_X(t-2)}{dt}\\
&= 1\times F_X'(t-2)\quad\text{on utilise ici }(u \circ v)' = v' \times (u' \circ v)\\
&= 1\times p_X(t-2)\\
p_Y(t) &= p_X(t-2)
\end{align}
La densité de \(Y\) est simplement un décalage de \(2\) vers la droite
de la densité de \(X\).
\hypertarget{b-loi-normale}{%
\subsubsection{(b) Loi normale}\label{b-loi-normale}}
Supposons que \(X\sim\mathcal{N}(0, \sigma_X^2=0.25)\). On a donc
\[p_X(t) = \frac{1}{\sqrt{2\pi\sigma_X^2}} \exp\left(-\frac{t^2}{2\sigma_X^2}\right).\]
\hypertarget{i-quelle-est-lexpression-de-p_y}{%
\paragraph{\texorpdfstring{(i) Quelle est l'expression de \(p_Y\)
?}{(i) Quelle est l'expression de p\_Y ?}}\label{i-quelle-est-lexpression-de-p_y}}
\[p_Y(t) = \frac{1}{\sqrt{2\pi\sigma_X^2}} \exp\left(-\frac{(t-2)^2}{2\sigma_X^2}\right).\]
\hypertarget{ii-donner-lallure-des-deux-densituxe9s-identifier-leur-maximum-et-les-valeurs-uxe0-123sigma-de-la-moyenne.}{%
\paragraph{\texorpdfstring{(ii) Donner l'allure des deux densités,
identifier leur maximum et les valeurs à \(\{1,2,3\}\sigma\) de la
moyenne.}{(ii) Donner l'allure des deux densités, identifier leur maximum et les valeurs à \textbackslash\{1,2,3\textbackslash\}\textbackslash sigma de la moyenne.}}\label{ii-donner-lallure-des-deux-densituxe9s-identifier-leur-maximum-et-les-valeurs-uxe0-123sigma-de-la-moyenne.}}
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{ \hspace*{\fill} \\}
\hypertarget{fonctions-mathrmerfcx-et-mathrmqx}{%
\section{\texorpdfstring{2. Fonctions \(\mathrm{erfc}(x)\) et
\(\mathrm{Q}(x)\)}{2. Fonctions \textbackslash mathrm\{erfc\}(x) et \textbackslash mathrm\{Q\}(x)}}\label{fonctions-mathrmerfcx-et-mathrmqx}}
Soit \(U \sim \mathcal{N}(0,1)\). La densité de \(U\) est donnée par
\[p_{U}(t)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{t^{2}}{2}\right).\]
\hypertarget{b-idem-avec-x-simmathcaln0sigma_x2.}{%
\subsection{\texorpdfstring{\#\#\# (b) Idem avec
\(X \sim\mathcal{N}(0,\sigma_{X}^2)\).}{\#\#\# (b) Idem avec X \textbackslash sim\textbackslash mathcal\{N\}(0,\textbackslash sigma\_\{X\}\^{}2).}}\label{b-idem-avec-x-simmathcaln0sigma_x2.}}
\hypertarget{a-probabilituxe9-derreur-mathbbpu-alpha}{%
\subsubsection{\texorpdfstring{(a) Probabilité d'erreur
\(\mathbb{P}(U > \alpha)\)}{(a) Probabilité d'erreur \textbackslash mathbb\{P\}(U \textgreater{} \textbackslash alpha)}}\label{a-probabilituxe9-derreur-mathbbpu-alpha}}
Les fonctions \(\mathrm{Q}\) et \(\mathrm{erfc}\) sont les suivantes:
\begin{align}
\operatorname{Q}(x) = \frac{1}{\sqrt{2 \pi}} \int_{x}^{+\infty} e^{\frac{-t^{2}}{2}} \mathrm{d} t \qquad\text{et}\qquad
\operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}} \int_{x}^{+\infty} e^{-t^{2}} \mathrm{d} t
\end{align}
Commencer par écrire \begin{align}
\mathbb{P}(U > \alpha) = \int_{\alpha}^{+\infty}{p_U(t) \mathrm{d}t} = \int_{\alpha}^{+\infty}{ \frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{t^{2}}{2}\right)\mathrm{d}t}
\end{align}
\hypertarget{a-probabilituxe9-derreur-mathbbpu-alpha-solution}{%
\subsubsection{\texorpdfstring{(a) Probabilité d'erreur
\(\mathbb{P}(U > \alpha)\)
(solution)}{(a) Probabilité d'erreur \textbackslash mathbb\{P\}(U \textgreater{} \textbackslash alpha) (solution)}}\label{a-probabilituxe9-derreur-mathbbpu-alpha-solution}}
\begin{itemize}
\tightlist
\item
\textbf{Avec \(\operatorname{Q}\) -} Le résultat est immédiat puisque
c'est la définition: \begin{align}
P(U>\alpha) = \frac{1}{\sqrt{2 \pi}} \int_{\alpha}^{+\infty} e^{\frac{-t^{2}}{2}} \mathrm{d} t
= \operatorname{Q}(\alpha)
\end{align}
\end{itemize}
%% \begin{center}\rule{0.5\linewidth}{\linethickness}\end{center}
\hypertarget{a-probabilituxe9-derreur-mathbbpu-alpha-solution}{%
\subsubsection{\texorpdfstring{(a) Probabilité d'erreur
\(\mathbb{P}(U > \alpha)\)
(solution)}{(a) Probabilité d'erreur \textbackslash mathbb\{P\}(U \textgreater{} \textbackslash alpha) (solution)}}\label{a-probabilituxe9-derreur-mathbbpu-alpha-solution}}
\begin{itemize}
\tightlist
\item
\textbf{Avec \(\operatorname{erfc}\) -} On va faire le changement de
variable suivant: \begin{align}
v = \frac{t}{\sqrt{2}} \Rightarrow v^2 = \frac{t^2}{2}\quad\text{et}\quad \mathrm{d}v = \frac{\mathrm{d}t}{\sqrt{2}}
\end{align} On obtient alors: \begin{align}
P(U>\alpha) = \frac{1}{\sqrt{2 \pi}} \int_{\alpha}^{+\infty} e^{\frac{-t^{2}}{2}} \mathrm{d} t
&= \frac{1}{\sqrt{2 \pi}} \int_{\frac{\alpha}{\sqrt{2}}}^{+\infty} e^{-v^2} \sqrt{2}\mathrm{d} v\\
&= \frac{1}{\sqrt{\pi}} \int_{\frac{\alpha}{\sqrt{2}}}^{+\infty} e^{-v^2} \mathrm{d} v\\
&= \frac{1}{2}\times\frac{2}{\sqrt{\pi}} \int_{\frac{\alpha}{\sqrt{2}}}^{+\infty} e^{-v^2} \mathrm{d} v\\
P(U>\alpha) &= \frac{1}{2}\operatorname{erfc}\left(\frac{\alpha}{\sqrt{2}}\right)
\end{align}
\end{itemize}
\hypertarget{b-probabilituxe9-derreur-mathbbpx-alpha}{%
\subsubsection{\texorpdfstring{(b) Probabilité d'erreur
\(\mathbb{P}(X > \alpha)\)}{(b) Probabilité d'erreur \textbackslash mathbb\{P\}(X \textgreater{} \textbackslash alpha)}}\label{b-probabilituxe9-derreur-mathbbpx-alpha}}
Cette fois-ci, on considère \(X\sim\mathcal{N}(0,\sigma_X^2)\).
Les fonctions \(\mathrm{Q}\) et \(\mathrm{erfc}\) sont les suivantes:
\begin{align}
\operatorname{Q}(x) = \frac{1}{\sqrt{2 \pi}} \int_{x}^{+\infty} e^{\frac{-t^{2}}{2}} \mathrm{d} t \qquad\text{et}\qquad
\operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}} \int_{x}^{+\infty} e^{-t^{2}} \mathrm{d} t
\end{align}
On peut commencer comme avec \(U\), ou \textbf{être un peu plus malin}.
Essayons de nous ramener au cas précédent.
Comment relier les fonctions de répartition de \(X\) et \(U\) ?
\begin{align}
\mathbb{P}(X > \alpha) = \mathbb{P}(U > \beta(\alpha))
\end{align}
\hypertarget{b-probabilituxe9-derreur-mathbbpu-alpha-solution}{%
\subsubsection{\texorpdfstring{(b) Probabilité d'erreur
\(\mathbb{P}(U > \alpha)\)
(solution)}{(b) Probabilité d'erreur \textbackslash mathbb\{P\}(U \textgreater{} \textbackslash alpha) (solution)}}\label{b-probabilituxe9-derreur-mathbbpu-alpha-solution}}
On posera \(U = \frac{X}{\sigma_X}\) qui a bien une variance de \(1\):
\begin{align}
\mathbb{V}[U] = \mathbb{V}\left[\frac{X}{\sigma_X}\right] = \frac{\mathbb{V}[X]}{\sigma_X^2} = \frac{\sigma_X^2}{\sigma_X^2} = 1
\end{align} et donc \begin{align}
\mathbb{P}(X > \alpha) = \mathbb{P}\left(U > \frac{\alpha}{\sigma_X}\right)
\end{align}
\begin{itemize}
\item ~
\hypertarget{avec-operatornameq---mathbbpleftu-fracalphasigma_xright-operatornameqleftfracalphasigma_xright.}{%
\subsection{\texorpdfstring{\textbf{Avec \(\operatorname{Q}\) -}
\(\mathbb{P}\left(U > \frac{\alpha}{\sigma_X}\right) = \operatorname{Q}\left(\frac{\alpha}{\sigma_X}\right)\).}{Avec \textbackslash operatorname\{Q\} - \textbackslash mathbb\{P\}\textbackslash left(U \textgreater{} \textbackslash frac\{\textbackslash alpha\}\{\textbackslash sigma\_X\}\textbackslash right) = \textbackslash operatorname\{Q\}\textbackslash left(\textbackslash frac\{\textbackslash alpha\}\{\textbackslash sigma\_X\}\textbackslash right).}}\label{avec-operatornameq---mathbbpleftu-fracalphasigma_xright-operatornameqleftfracalphasigma_xright.}}
\item
\textbf{Avec \(\operatorname{erfc}\) -}
\(\mathbb{P}\left(U > \frac{\alpha}{\sigma_X}\right) = \frac{1}{2}\operatorname{erfc}\left(\frac{\alpha}{\sqrt{2}\sigma_X}\right)\).
\end{itemize}
\hypertarget{application}{%
\section{3. Application}\label{application}}
On considère une transmission de type BPSK:
\begin{longtable}[]{@{}cc@{}}
\toprule
Bit \(-B\) & Amplitude émise \(-E\)\tabularnewline
\midrule
\endhead
\(1\) & \(2 V\)\tabularnewline
\(0\) & \(0 V\)\tabularnewline
\bottomrule
\end{longtable}
Le canal de communication est bruité.
Il \textbf{ajoute} au signal transmis \(E\) une VAR
\(N\sim\mathcal{N}(0,\sigma^2)\).
On reçoit donc la variable aléatoire \(Y\) définit par: \begin{align}
Y = E + N
\end{align}
\hypertarget{a-loi-de-y}{%
\subsubsection{\texorpdfstring{(a) Loi de
\(Y\)}{(a) Loi de Y}}\label{a-loi-de-y}}
Rappelons que
\[\mathbb{E}[Y] = \mathbb{E}[E + N] = \mathbb{E}[E] + \mathbb{E}[N] = E + \mathbb{E}[N]\]
et (\(E\) et \(N\) étant indépendents)
\[\mathbb{V}[Y] = \mathbb{V}[E + N] = \mathbb{V}[E] + \mathbb{V}[N] = \mathbb{V}[N]\]
Quelle est la loi de \(Y\) selon le bit émis ?
% \begin{center}\rule{0.5\linewidth}{\linethickness}\end{center}
\textbf{Réponse:}
\begin{longtable}[]{@{}ccc@{}}
\toprule
\(B\) & \(E\) & \(Y\)\tabularnewline
\midrule
\endhead
\(1\) & \(2 V\) & \(\mathcal{N}(2,\sigma^2)\)\tabularnewline
\(0\) & \(0 V\) & \(\mathcal{N}(0,\sigma^2)\)\tabularnewline
\bottomrule
\end{longtable}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{3}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{from} \PY{n+nn}{ipywidgets} \PY{k+kn}{import} \PY{o}{*}
\PY{k+kn}{import} \PY{n+nn}{ipywidgets} \PY{k}{as} \PY{n+nn}{widgets}
\PY{k}{def} \PY{n+nf}{pdf\PYZus{}plot}\PY{p}{(}\PY{n}{mu\PYZus{}0}\PY{p}{,} \PY{n}{mu\PYZus{}1}\PY{p}{)}\PY{p}{:}
\PY{c+c1}{\PYZsh{} Calcules les densités de probabilité pour chaque variable}
\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mf}{3.0}\PY{p}{,} \PY{l+m+mf}{6.0}\PY{p}{,} \PY{n}{num}\PY{o}{=}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{;}
\PY{n}{y\PYZus{}0} \PY{o}{=} \PY{n}{normpdf}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{mu\PYZus{}0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} \PY{c+c1}{\PYZsh{} densité de probabilité de X}
\PY{n}{y\PYZus{}1} \PY{o}{=} \PY{n}{normpdf}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{mu\PYZus{}1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} \PY{c+c1}{\PYZsh{} densité de probabilité de X + 2}
\PY{c+c1}{\PYZsh{} Trace les courbes}
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\PY{n}{ax}\PY{o}{=}\PY{n}{plt}\PY{o}{.}\PY{n}{axes}\PY{p}{(}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{grid}\PY{p}{(}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{y\PYZus{}0}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}p\PYZus{}}\PY{l+s+s2}{\PYZob{}}\PY{l+s+s2}{Y|B\PYZcb{}(t|0)\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{y\PYZus{}1}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}p\PYZus{}}\PY{l+s+s2}{\PYZob{}}\PY{l+s+s2}{Y|B\PYZcb{}(t|1)\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{;}
\PY{c+c1}{\PYZsh{} Trace les zones d\PYZsq{}erreurs}
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\PY{n}{ax}\PY{o}{.}\PY{n}{fill\PYZus{}between}\PY{p}{(}\PY{n}{x}\PY{p}{[}\PY{n}{y\PYZus{}0} \PY{o}{\PYZlt{}} \PY{n}{y\PYZus{}1}\PY{p}{]}\PY{p}{,} \PY{n}{y\PYZus{}0}\PY{p}{[}\PY{n}{y\PYZus{}0} \PY{o}{\PYZlt{}} \PY{n}{y\PYZus{}1}\PY{p}{]}\PY{p}{,} \PY{n}{facecolor}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{blue}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{alpha}\PY{o}{=}\PY{o}{.}\PY{l+m+mi}{5}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{hat}\PY{l+s+si}{\PYZob{}B\PYZcb{}}\PY{l+s+s2}{=0 }\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mid B=1\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Paramètrage de la figure}
\PY{n}{ax}\PY{o}{.}\PY{n}{set\PYZus{}xlabel}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}y\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{set\PYZus{}ylabel}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}p\PYZus{}}\PY{l+s+s2}{\PYZob{}}\PY{l+s+s2}{Y|B\PYZcb{}(t}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mid b)\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{)}\PY{p}{;}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}\PY{p}{;}
\end{Verbatim}
\end{tcolorbox}
\hypertarget{b-probabilituxe9-derreur}{%
\subsubsection{(b) Probabilité
d'erreur}\label{b-probabilituxe9-derreur}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{4}{\boxspacing}
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\PY{n}{interact}\PY{p}{(}\PY{n}{pdf\PYZus{}plot}\PY{p}{,}
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\PY{n}{mu\PYZus{}1}\PY{o}{=}\PY{n}{widgets}\PY{o}{.}\PY{n}{FloatSlider}\PY{p}{(}\PY{n}{value}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{,}\PY{n+nb}{min}\PY{o}{=}\PY{l+m+mf}{1.0}\PY{p}{,}\PY{n+nb}{max}\PY{o}{=}\PY{l+m+mf}{3.0}\PY{p}{,}\PY{n}{step}\PY{o}{=}\PY{l+m+mf}{0.1}\PY{p}{)}\PY{p}{,}
\PY{p}{)}\PY{p}{;}
\end{Verbatim}
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\begin{verbatim}
interactive(children=(FloatSlider(value=0.0, description='mu_0', max=1.0, min=-1.0), FloatSlider(value=2.0, de…
\end{verbatim}
\hypertarget{c-valeurs-de-bruit-causant-une-erreur}{%
\subsubsection{(c) Valeurs de bruit causant une
erreur}\label{c-valeurs-de-bruit-causant-une-erreur}}
D'après la figure précédente, on voit qu'il y a une erreur lorsque: - Si
\(B=0\): \(e_0 + N > \lambda \Rightarrow N > \lambda - e_0\) - Si
\(B=1\): \(e_1 + N < \lambda \Rightarrow N < \lambda - e_1\)
\hypertarget{d-probabilituxe9-derreur-binaire}{%
\subsubsection{(d) Probabilité d'erreur
binaire}\label{d-probabilituxe9-derreur-binaire}}
Pour calculer la probabilité d'erreur binaire, on effectuera les étapes
suivantes:
\begin{itemize}
\item
\textbf{Proba. totales et Bayes}
\item
\textbf{Faire apparaître \(N\)}
\item
\textbf{Utiliser les propriétés de la gaussienne}
\end{itemize}
\hypertarget{d-probabilituxe9-derreur-binaire}{%
\subsubsection{(d) Probabilité d'erreur
binaire}\label{d-probabilituxe9-derreur-binaire}}
Pour calculer la probabilité d'erreur binaire, on effectuera les étapes
suivantes:
\begin{itemize}
\item
\textbf{Proba. totales et Bayes} \begin{align}
P_e &= \mathbb{P}(B=0 \cap Y>\lambda)+\mathbb{P}(B=1 \cap Y<\lambda)&\text{(formule des probabilités totales})\\
&= \mathbb{P}(Y>\lambda \mid B=0) \mathbb{P}(B=0)+\mathbb{P}(Y<\lambda \mid B=1) \mathrm{P}(B=1) &\text{(formule de Bayes})\\
&= \mathbb{P}(Y>\lambda \mid E=e_0) \mathbb{P}(B=0)+\mathbb{P}(Y<\lambda \mid E=e_1) \mathrm{P}(B=1)\\
&= \frac{1}{2}\left(\mathbb{P}\left(Y>\lambda \mid E=e_0\right) + \mathbb{P}\left(Y<\lambda \mid E=e_1\right)\right)
\end{align}
\item
\emph{Faire apparaître \(N\)}
\item
\emph{Utiliser les propriétés de la gaussienne}
\end{itemize}
\hypertarget{d-probabilituxe9-derreur-binaire}{%
\subsubsection{(d) Probabilité d'erreur
binaire}\label{d-probabilituxe9-derreur-binaire}}
Pour calculer la probabilité d'erreur binaire, on effectuera les étapes
suivantes:
\begin{itemize}
\item
\emph{Proba. totales et Bayes} \begin{align}
P_e &= \frac{1}{2}\left(\mathbb{P}\left(Y>\lambda \mid E=e_0\right) + \mathbb{P}\left(Y<\lambda \mid E=e_1\right)\right)
\end{align}
\item
\textbf{Faire apparaître \(N\)} \begin{align}
P_e &= \frac{1}{2}\left(\mathbb{P}\left(e_0 + N>\lambda \mid E=e_0\right) + \mathbb{P}\left(e_1 + N<\lambda \mid E=e_1\right)\right)\\
&= \frac{1}{2}\left(\mathbb{P}\left(N>\lambda-e_0 \mid E=e_0\right) + \mathbb{P}\left(N<\lambda-e_1 \mid E=e_1\right)\right)\\
&= \frac{1}{2}\left(\mathbb{P}\left(N>\frac{e_1-e_0}{2} \mid E=e_0\right) + \mathbb{P}\left(N<\frac{e_0-e_1}{2} \mid E=e_1\right)\right)
\end{align}
\item
\emph{Utiliser les propriétés de la gaussienne}
\end{itemize}
\hypertarget{d-probabilituxe9-derreur-binaire}{%
\subsubsection{(d) Probabilité d'erreur
binaire}\label{d-probabilituxe9-derreur-binaire}}
Pour calculer la probabilité d'erreur binaire, on effectuera les étapes
suivantes:
\begin{itemize}
\item
\emph{Proba. totales et Bayes} \begin{align}
P_e &= \mathbb{P}(Y>\lambda \mid E=e_0) \mathbb{P}(B=0)+\mathbb{P}(Y<\lambda \mid E=e_1) \mathrm{P}(B=1)
\end{align}
\item
\emph{Faire apparaître \(N\)} \begin{align}
P_e &= \frac{1}{2}\left(\mathbb{P}\left(N>\frac{e_1-e_0}{2} \mid E=e_0\right) + \mathbb{P}\left(N<\frac{e_0-e_1}{2} \mid E=e_1\right)\right)
\end{align}
\item
\textbf{Utiliser les propriétés de la gaussienne} \begin{align}
P_e &= \frac{1}{2}\left(\mathbb{P}\left(N>\frac{e_1-e_0}{2} \mid E=e_0\right) + \mathbb{P}\left(N<-\frac{e_1-e_0}{2} \mid E=e_1\right)\right)\\
&= \frac{1}{2}\left(\mathbb{P}\left(N>\frac{e_1-e_0}{2} \mid E=e_0\right) + \mathbb{P}\left(N>\frac{e_1-e_0}{2} \mid E=e_1\right)\right) &\text{(la gaussienne est symétrique)}\\
&= \mathbb{P}\left(N>\frac{e_1-e_0}{2} \mid E=e_0\right)\\
&= \operatorname{Q}\left(\frac{e_1-e_0}{2\sigma}\right)&\text{(d'après la question 2.(b))}
\end{align}
\end{itemize}
\hypertarget{d-probabilituxe9-derreur-binaire}{%
\subsubsection{(d) Probabilité d'erreur
binaire}\label{d-probabilituxe9-derreur-binaire}}
Pour calculer la probabilité d'erreur binaire, on effectuera les étapes
suivantes:
\begin{itemize}
\item
\emph{Proba. totales et Bayes} \begin{align}
P_e &= \mathbb{P}(Y>\lambda \mid E=e_0) \mathbb{P}(B=0)+\mathbb{P}(Y<\lambda \mid E=e_1) \mathrm{P}(B=1)
\end{align}
\item
\emph{Faire apparaître \(N\)} \begin{align}
P_e &= \frac{1}{2}\left(\mathbb{P}\left(N>\frac{e_1-e_0}{2} \mid E=e_0\right) + \mathbb{P}\left(N<\frac{e_0-e_1}{2} \mid E=e_1\right)\right)
\end{align}
\item
\emph{Utiliser les propriétés de la gaussienne} \begin{align}
P_e &= \operatorname{Q}\left(\frac{e_1-e_0}{2\sigma}\right)
\end{align}
\end{itemize}
\hypertarget{on-obtient-un-ruxe9sultat-simple-qui-duxe9pend-directement-de-luxe9cart-e_1---e_0}{%
\paragraph{\texorpdfstring{On obtient un résultat simple qui dépend
directement de l'écart \(e_1 - e_0\)
!}{On obtient un résultat simple qui dépend directement de l'écart e\_1 - e\_0 !}}\label{on-obtient-un-ruxe9sultat-simple-qui-duxe9pend-directement-de-luxe9cart-e_1---e_0}}
\hypertarget{e-bit-error-rate-ber-et-snr}{%
\subsubsection{(e) Bit Error Rate (BER) et
SNR}\label{e-bit-error-rate-ber-et-snr}}
On admet que la puissance du signal et celle du bruit sont égales à
\begin{align}
P_S = \frac{e_0^2 + e_1^2}{2}\quad\text{et}\quad P_N = \sigma^2
\end{align} Le SNR vaut donc \begin{align}
\mathsf{SNR} = \frac{P_S}{\sigma^2}
\end{align}
\begin{itemize}
\tightlist
\item
\textbf{Si \(e_0 = 0\):} \(P_S = \frac{e_1^2}{2}\) et
\(\mathsf{SNR} = \frac{e_1^2}{2\sigma^2} \Rightarrow P_e = \mathrm{Q}\left(\sqrt{\frac{\mathrm{SNR}}{2}}\right)\)
\item
\textbf{Si \(e_0 = -e_1\):} \(P_S = e_1^2\) et
\(\mathsf{SNR} = \frac{e_1^2}{\sigma^2} \Rightarrow P_e = \mathrm{Q}\left(\sqrt{\mathrm{SNR}}\right)\)
\end{itemize}
\hypertarget{bonus---tracuxe9-de-p_e}{%
\section{\texorpdfstring{Bonus - tracé de
\(P_e\)}{Bonus - tracé de P\_e}}\label{bonus---tracuxe9-de-p_e}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{5}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{stats} \PY{k+kn}{import} \PY{n}{norm}
\PY{k}{def} \PY{n+nf}{plot\PYZus{}BER}\PY{p}{(}\PY{n}{snr}\PY{p}{)}\PY{p}{:}
\PY{c+c1}{\PYZsh{} Generate a normal random variable}
\PY{n}{rv} \PY{o}{=} \PY{n}{norm}\PY{p}{(}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Trace les courbes}
\PY{n}{fig} \PY{o}{=} \PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{n}{figsize}\PY{o}{=}\PY{p}{(}\PY{l+m+mi}{16}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{)}\PY{p}{,} \PY{n}{dpi}\PY{o}{=}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{=}\PY{n}{plt}\PY{o}{.}\PY{n}{axes}\PY{p}{(}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{grid}\PY{p}{(}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{semilogy}\PY{p}{(}\PY{n}{snr}\PY{p}{,} \PY{n}{rv}\PY{o}{.}\PY{n}{pdf}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{sqrt}\PY{p}{(}\PY{n}{snr}\PY{o}{/}\PY{l+m+mi}{2}\PY{p}{)}\PY{p}{)}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{k\PYZhy{}\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathrm}\PY{l+s+si}{\PYZob{}Q\PYZcb{}}\PY{l+s+s2}{(}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{sqrt}\PY{l+s+s2}{\PYZob{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathsf}\PY{l+s+si}{\PYZob{}SNR\PYZcb{}}\PY{l+s+s2}{/2\PYZcb{})\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{semilogy}\PY{p}{(}\PY{n}{snr}\PY{p}{,} \PY{n}{rv}\PY{o}{.}\PY{n}{pdf}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{sqrt}\PY{p}{(}\PY{n}{snr}\PY{p}{)}\PY{p}{)}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{k\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathrm}\PY{l+s+si}{\PYZob{}Q\PYZcb{}}\PY{l+s+s2}{(}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{sqrt}\PY{l+s+s2}{\PYZob{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathsf}\PY{l+s+si}{\PYZob{}SNR\PYZcb{}}\PY{l+s+s2}{\PYZcb{})\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Paramètrage de la figure}
\PY{n}{ax}\PY{o}{.}\PY{n}{set\PYZus{}xlabel}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathsf}\PY{l+s+si}{\PYZob{}SNR\PYZcb{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{set\PYZus{}ylabel}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathrm}\PY{l+s+si}{\PYZob{}Q\PYZcb{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{;}
\PY{n}{ax}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{)}\PY{p}{;}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}\PY{p}{;}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{6}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{plot\PYZus{}BER}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{20}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_34_0.png}
\end{center}
{ \hspace*{\fill} \\}
% Add a bibliography block to the postdoc
\end{document}