From d3f89a314892e005fe27a77b89e50cae27b6e81c Mon Sep 17 00:00:00 2001 From: Pierre-antoine Comby Date: Thu, 2 May 2019 17:22:44 +0200 Subject: [PATCH] =?UTF-8?q?d=C3=A9but=20de=20la=20refonte=20de=20TD=20455?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 455-Codage_Sources/TD/TD1/TD1.tex | 241 --------------------------- 455-Codage_Sources/TD/TD1/arbre1.png | Bin 41061 -> 0 bytes 455-Codage_Sources/TD/TD1/arbre2.png | Bin 49159 -> 0 bytes 455-Codage_Sources/TD/TD1/arbre3.png | Bin 21167 -> 0 bytes 455-Codage_Sources/TD/TD1/arbre4.png | Bin 75285 -> 0 bytes 455-Codage_Sources/TD/TD2/TD2.tex | 91 ---------- 455-Codage_Sources/TD/TD4/TD4.tex | 133 --------------- 455-Codage_Sources/TD/TD5/1.png | Bin 17770 -> 0 bytes 455-Codage_Sources/TD/TD5/10.png | Bin 19975 -> 0 bytes 455-Codage_Sources/TD/TD5/11.png | Bin 21568 -> 0 bytes 455-Codage_Sources/TD/TD5/2.png | Bin 38924 -> 0 bytes 455-Codage_Sources/TD/TD5/3.png | Bin 25408 -> 0 bytes 455-Codage_Sources/TD/TD5/4.png | Bin 23973 -> 0 bytes 455-Codage_Sources/TD/TD5/5.png | Bin 14143 -> 0 bytes 455-Codage_Sources/TD/TD5/6.png | Bin 19569 -> 0 bytes 455-Codage_Sources/TD/TD5/7.png | Bin 12893 -> 0 bytes 455-Codage_Sources/TD/TD5/8.png | Bin 23843 -> 0 bytes 455-Codage_Sources/TD/TD5/9.png | Bin 16776 -> 0 bytes 455-Codage_Sources/TD/TD5/TD5.tex | 239 -------------------------- 455-Codage_Sources/TD/TD6/1.png | Bin 33316 -> 0 bytes 455-Codage_Sources/TD/TD6/2.png | Bin 19997 -> 0 bytes 455-Codage_Sources/TD/TD6/3.png | Bin 65982 -> 0 bytes 455-Codage_Sources/TD/TD6/TD6.tex | 212 ----------------------- 455-Codage_Sources/TD/TD7/TD7.tex | 156 ----------------- 24 files changed, 1072 deletions(-) delete mode 100644 455-Codage_Sources/TD/TD1/TD1.tex delete mode 100644 455-Codage_Sources/TD/TD1/arbre1.png delete mode 100644 455-Codage_Sources/TD/TD1/arbre2.png delete mode 100644 455-Codage_Sources/TD/TD1/arbre3.png delete mode 100644 455-Codage_Sources/TD/TD1/arbre4.png delete mode 100644 455-Codage_Sources/TD/TD2/TD2.tex delete mode 100644 455-Codage_Sources/TD/TD4/TD4.tex delete mode 100644 455-Codage_Sources/TD/TD5/1.png delete mode 100644 455-Codage_Sources/TD/TD5/10.png delete mode 100644 455-Codage_Sources/TD/TD5/11.png delete mode 100644 455-Codage_Sources/TD/TD5/2.png delete mode 100644 455-Codage_Sources/TD/TD5/3.png delete mode 100644 455-Codage_Sources/TD/TD5/4.png delete mode 100644 455-Codage_Sources/TD/TD5/5.png delete mode 100644 455-Codage_Sources/TD/TD5/6.png delete mode 100644 455-Codage_Sources/TD/TD5/7.png delete mode 100644 455-Codage_Sources/TD/TD5/8.png delete mode 100644 455-Codage_Sources/TD/TD5/9.png delete mode 100644 455-Codage_Sources/TD/TD5/TD5.tex delete mode 100644 455-Codage_Sources/TD/TD6/1.png delete mode 100644 455-Codage_Sources/TD/TD6/2.png delete mode 100644 455-Codage_Sources/TD/TD6/3.png delete mode 100644 455-Codage_Sources/TD/TD6/TD6.tex delete mode 100644 455-Codage_Sources/TD/TD7/TD7.tex diff --git a/455-Codage_Sources/TD/TD1/TD1.tex b/455-Codage_Sources/TD/TD1/TD1.tex deleted file mode 100644 index 339ae97..0000000 --- a/455-Codage_Sources/TD/TD1/TD1.tex +++ /dev/null @@ -1,241 +0,0 @@ -\documentclass{article} -\input{../../preambule/preambule} - -\newcommand{\nom}{TD1 : Codage de Sources} -\renewcommand{\nomentete}{UE455 - \nom} - - - -\begin{document} - -\titre{\nom} - -\begin{center} -TD et TP : remi.azouit@ens-cachan.fr -\end{center} - -\section*{Point de cours sur le codage de Huffman} - -Il s'agit d'un codage préfixe construit de façon récursive. Avec un codage préfixe, on n'a pas besoin de recevoir tout pour décoder. On peut décoder au fur et à mesure de la réception. Dans un code préfixe, chaque code ne peut pas être le début d'un autre code. - -\begin{exemple} - -On considère une source $X$ à valeurs dans $\X=\{A,B,...G\}$. - -On a $p_A=3/8,\quad p_B=3/16, \quad p_C=3/16, \quad p_D=1/8,\quad p_E=1/16,\quad p_F=1/32,\quad p_G=1/32$.\\ - -Pour construire le codage, on range les symboles par probabilité d'apparition décroissante et on construit un arbre en associant à chaque fois les deux probabilités les plus faibles. Ainsi, les symboles les plus probables ont les codes les plus courts. Si on a deux probabilités qui sont égales, on peut choisir de façon équivalente l'une des deux : le codage sera différent mais tout aussi efficace. - -\begin{figure}[h!] -\centering -\includegraphics[scale=0.5]{arbre1.png} -\end{figure} - -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|c|c|} -\hline -$X$ & A & B & C & D & E & F & G \\ -\hline -Code & 1 & 011 & 010 & 001 & 0001 & 00001 & 00000 \\ -\hline -\end{tabular} -\end{center} - -\begin{rmq} -Pour coder un texte, on peut soit utiliser des probabilités et donc des codes "généraux", ou alors les recalculer à chaque fois. -\end{rmq} -\end{exemple} - -\newpage -\section*{Exercice 1 : Codage de Huffman} - -\begin{enumerate} -\item Construction de l'arbre puis codage - -\begin{figure}[h!] -\centering -\includegraphics[scale=0.5]{arbre2.png} -\end{figure} - -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|c|c|c|} -\hline -X & D & B & A & F & E & C & G & H \\ -\hline -Code & 00 & 10 & 010 & 110 & 111 & 0110 & 01110 & 01111 \\ -\hline -\end{tabular} -\end{center} - -\item $R = \sum_{x\in\X}p_x \times l_x$ où $l_x$ est la longueur du code associé à $x$ - -Ainsi, $R=2.83$ bits/symbole. - -\item L'entropie est le nombre minimal de bits par symbole nécessaire pour représenter la source: - -$H=-\sum_{x\in\X} p_x \log_2 p_x$ - -Ainsi $H=2.7978$ bits/symbole.\\ - -On a bien $H \leq R$. Notre codage n'est pas le codage optimal. - -\begin{rmq} -On obtiendrait $R=H$ dans le cas où toutes les probabilités étaient des puissances de 2. -\end{rmq} - -\item Le code correspondant à $ABAADE$ est : -$010|10|010|010|00|111$ -\end{enumerate} - -\newpage -\section*{Exercice 2 : Codage d'une source de Markov} - -\begin{enumerate} -\item Le nombre minimal de bits par symbole est déterminé par l'entropie. Le code le plus efficace aurait donc un nombre moyen de bits par symbole égal à l'entropie $H=\sum p_i \log(\frac{1}{p_i}) = 1.19$ bits/symbole. - -\item Construction de l'arbre puis du codage - -\begin{figure}[h!] -\centering -\includegraphics[scale=0.5]{arbre3.png} -\end{figure} - -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline -X & ADD & STO & SUB & MPY \\ -\hline -\hline -CODE & 0 & 10 & 110 & 111 \\ -\hline -\end{tabular} -\bigbreak -\end{center} -On calcule alors le débit moyen R = $\sum_{x \in \X} pr(x).Nbbit(X)$ = 1.375 bit/symbole\\ - -On a une valeur supérieure à l'entropie mais inférieure à la valeur initiale. Cool! Notre codage n'est pas optimal mais est meilleur que le premier proposé.\\ - - -On considère maintenant le modèle de Markov d'ordre 1 pour modéliser la succession d'instructions constituany un programme. La matrice de transmission de ce modèle est donnée par: -\[P_{j|i} = \begin{pmatrix} -\frac{13}{16}&\frac{1}{16}&\frac{1}{16}&\frac{1}{16}\\ -0&0&0&1\\ -\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\\ -1&0&0&0 -\end{pmatrix}\] -\begin{enumerate} -\item Montrer que $p = \left(\frac{3}{4}\frac{1}{16},\frac{1}{16}, \frac{1}{8} \right)$ représente bien la distribution de probabilité stationnaire de la source de Markov d'ordre 1.\\ -Il s'agit de vérifier que $p\times P_{j|i} = p$.\\ - -\underline{Remarque}: On peut aussi voir ce résultat sous la forme plus courante $p^T = P_{j|i}^T p^T$. Ceci permet notamment de calculer $p$ s'il n'est pas donné, en cherchant un vecteur propre de $P_{j|i}^T$.\\ - -\item Calculer les probabilités moyennes d'avoir une succession des deux instructions suivantes :\\ - -\underline{Remarque}: ne pas confondre les lignes et les colonnes dans $P_{j|i}$: -\[P_{j|i} = \begin{pmatrix} -p_{ADD|ADD} & p_{SUB|ADD} & p_{MPY|ADD} & p_{STO|ADD}\\ -p_{ADD|SUB} & p_{SUB|SUB} & p_{MPY|SUB} & \vdots \\ -\vdots & \dots & \dots & \vdots\\ -p_{ADD|STO} & \dots & \dots & p_{STO|STO} -\end{pmatrix}\] -On a donc avec la règle de Bayes : -\begin{align*} -p(STO - ADD) &= p(ADD_j | STO_{j-1}).p(STO_{j-1})\\ -&= 1 \times \frac{1}{8}\\ -&= \frac{1}{8}\\ -p(ADD - ADD) &= \frac{13}{16}\times \frac{3}{4}\\ -&= \frac{39}{64}\\ -p(SUB-MPY) &= 0 \times \frac{1}{16}\\ -&= 0 -\end{align*} -On peut calculer les autres probabilités cumulées de la même façon à partir des probabilités conditionnelles en les piochant dans le tableau.(Elles seront toutes données à la question 4.)\\ - -\item En utilisant le modèle de Markov d'ordre 1, dans quelles proportions est-il possible de réduire le nombre moyen de bits utilisés pour coder une instruction?\\ - -On calcule donc l'entropie $H=0.87$bit/symbole. On a une entropie inférieur à l'entropie initiale. Ceci vient du fait que l'on a un couplage des symboles, donc plus d'informations pour un même nombre de bits transmis.\\ -\end{enumerate} - -\item -On commence par rappeler toutes les probabilités cumulées ordonnées:\\ -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|} -\hline -X & ADD-ADD & STO-ADD & SUB-STO & ADD-SUB & ADD-MPY \\ -\hline -Pr(X) & $\frac{39}{64}$ & $\frac{8}{64}$ & $\frac{4}{64}$ & $\frac{3}{64}$ & $\frac{3}{64}$ \\ -\hline -\hline -X & ADD-STO & MPY-ADD & MPY-SUB & MPY-MPY & MPY-STO \\ -\hline -Pr(X) & $\frac{3}{64}$ & $\frac{1}{64}$ & $\frac{1}{64}$ & $\frac{1}{64}$ & $\frac{1}{64}$ \\ -\hline -\end{tabular} -\end{center} -\bigbreak -On a alors $R = \frac{1.98}{2}$ bit/instruction. (On divise par 2 car on a 2 instructions codées à chaque fois) -\end{enumerate} - -\newpage -\section*{Exercice 14 : Codage de source distribué} - -\begin{enumerate} - -\item On a 8 possibilités et toutes équiprobables donc : -\begin{align*} -H(X) &= -\sum p_i log_2 p_i\\ -&= -8 (\frac{1}{8} log_2 \frac{1}{8})\\ -H(X) &= 3 bit/symbole = H(Y) -\end{align*} - -\item On a $d_H(x_k,y_k) \leq 1$ ce qui impose certaines combinaisons. Pour un x donné, il y a 4 possibilités de y. On a donc $8\times 4 =32$ possibilités toutes équiprobables.\\ -Ainsi, $H(x_k,y_k) = 5$ bit/symbole.\\ - -\item On a par propriété : $H(X,Y) = H(X|Y) + H(Y)$, or si X et Y sont indépendants alors $H(X|Y) = H(X)$ donc on a $H(X|Y) = H(X) + H(Y)$.\\ -Comme on n'a pas cette égalité, c'est que X et Y sont corrélés.\\ - -\item Il faut 3 bits pour coder $x_k$\\ -%Trou du cul!!! -\item $H(X,Y) = H(X|Y) + H(Y)$ donc on a clairement $H(X|Y)=2$.\\ - -Comme $d_H(x_k,y_k) \leq 1$, on a qu'un seul bite de différent. Il suffit donc de coder la position du bite différent, 00 si identique, 01 pour le premier etc.\\ - -\item Pour coder le couple, il ne faut plus que 5 bits.\\ - -\item Dans ce cas, s'ils ne communiquent pas, on va devoir coder avec 6 bits.\\ - -\item 4 ensembles, donc on n'a besoin que de deux bits.\\ - -\item On a donc besoin que de 5 bits.\\ - -\item On connaît x et l'index de y, et x a un bit de différent avec l'une des deux possibilités de l'ensemble correspondant à l'index donné par y. Ces 2 possibilités sont distantes de 3, donc x ne peut être distant de 1 qu'avec l'un des deux. On peut donc retrouver (seulement car $d_H(x,y)\leq 1$) y à partir de x.\\ - -\item On alors R=H !!! \huge youuuupiiiiiiiiii \small -\end{enumerate} - -\newpage -\section*{Exercice 13 : Codage par plages de zéros} -\begin{enumerate} -\item On calcule l'entropie de cette source: $H(X) = -(p log(p) + (1-p)log(1-p))$\\ - -\item Huffman serait inefficace (inutile) car il y a seulement 2 symboles à coder.\\ - -\item On a $y \in [0;+\infty [$.\\ - -\item On a $P_0 = p$, $P_1 = p(1-p)$, $P_2 = p(1-p)^2$ et $P_k = p(1-p)^k$.\\ - -\item On a $H(Y) = -\sum_{k=0}^{\infty} P_k logP_k$.\\ - -\item \begin{align*} -H(Y) &= - \sum_{k=0}^{\infty}p(1-p)^klog(p(1-p)^k)\\ -&= - \sum_{k=0}^{\infty}p(1-p)^k(log(p)+klog(1-p))\\ -&= -log(p) - \frac{1-p}{p} log(1-p) -\end{align*} - -\item On part de $-log(p_k) \leq \lceil-log(p_k) \rceil \leq -log(p_k) + 1$ en multipliant par $p_k$ et n sommant sur l'infini, on identifie l'entropie et on obtient: -\[H(Y) \leq l(Y) \leq H(Y) +1 \] - -\item k pour Y représente 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zke)!Q&AN3B1Y!D59){DTN_Zi2EjZiN9~vtq-#TkQU#`6~9wQTUd6zCbMeHvU86b0_U+$^nULA`qkoE zLPUtds5CiH~`exV-Ta1y|+vH|E5gXAy zz*-Bu^3d#W$xkqt@gNtAKc%h;5vwrUd$0q+5;cLh_d-LBE0-=&1zV1?{i~^i1QLxxjYDNCS;d~VR`GfOC?3l(VJ6>gtlDC0o$M3-1z3TRHF6A!-q zyU7yfu7(ZZ?~+sourt2CpM`om-AUm=d8Ad|aNWR2D~k!7jK&Fl*ieD;@B{(^hR$zh z3zk}CaJ*8jE6G#kB4O3YkVvqmgE@pPd_#qM`q$5CebW+`>xtAAg_os*(}=2@ldkUx zF{%GzJW}tNRKg-f6P!A5jNM!_J5Vb{-<@ z|Ga#{uxNI!*S;TBN@O2N=3bU-~RvqE5-BZ5O(Y^6$o@w z;P7$F&CLUM|2Z5X>a-*}-f^qAQf29D%_N(C7n@zHR4Xuq^%IY%MOdjSNhd?tS;N0_lG;7)2pf@c z?#vI|GFG$.\\ -Si $\omega_i = 0$ on code $ <0, 0, \xi_{(i)}>$. - -ET on oublie pas : $c(i,:) = $, c'est à dire: ??!?. Ça traine tout seul dans son coin du tableau. - -\item Pour coder la chaîne bongbonbongobang$\$$, on commence par coder les quatre premiers symboles, c'est à dire que l'on a: -\[<0,0 <0,0,o> <0,0,n> <0,0,g>\] -Puis on code les quatre suivant ensemble avec $<1,3,b>$ car la sous-chaine composé de $bon$ apparait déjà avant.\\ -Puis $ongo$ car la racine $ong$ existe déjà donc on a $<2,3,o>$.\\ -Ensuite on code $ba$ qui donne $<1,1,a>$.\\ -Et pour finir on code $ng\$$ qui donne $<3,2,\$>$. - - -\item \begin{lstlisting} -c; //vecteur de triples -x = []; // séquence décodée -for i=1:length(c) - x = [x, x[c(i,1):c(i,1)+c(i,2)], c(i,3)] -end -\end{lstlisting} - -\item Pour coder la chaîne bongbonbongobang$\$$, on commence par coder les quatre premiers symboles, c'est à dire que l'on a: -\[<0,0 <0,0,o> <0,0,n> <0,0,g>\] -Puis on code les quatre suivant ensemble avec $<1,3,b>$ car la sous-chaine composé de $bon$ apparait déjà avant.\\ -Puis $ongo$ car la racine $ong$ existe déjà donc on a $<2,3,o>$.\\ -Ensuite on code $ba$ qui donne $<1,1,a>$.\\ -Et pour finir on code $ng\$$ qui donne $<3,2,\$>$. - -\item On effectue le même travail en décodant cette fois. Si on ne se trompe pas, on retombe sur la même chose. - -\item Avec la méthode C, pour coder bongbongbonbongobang$\$$ on obtient: -$<0,0,b> ... <0,0,g> <1,7,b><2,3,o><1,1,a><3,2,\$>$ -Ce que l'on retient c'est que pour bongbon on a une répétition sur 7 symbole des 4 premiers, après il suffit d'appliquer bêtement l'énoncé. - -\item \begin{lstlisting} -c; -x=[]; -for i=1:length(c) - for j = 1:c(i,2) - x = x[x,x[c(i,1+j-1)]; - end -end -\end{lstlisting} - -LE problème est que ? pas compris ce qu'il a dit. - -\item \begin{align*} -0 \leq j \leq s \Rightarrow \lceillog_2 S \rceil \text{ bits} \\ -0 \leq k \leq s + t \Rightarrow \lceillog_2 (S+t) \rceil \text{ bits} \\ -u \Rightarrow \lceillog_2 u \rceil \text{ bits} \\ -\end{align*} - - - - - -\end{enumerate} - - - -\end{document} \ No newline at end of file diff --git a/455-Codage_Sources/TD/TD5/1.png b/455-Codage_Sources/TD/TD5/1.png deleted file mode 100644 index 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\Rightarrow D_1 = \frac{\Delta_2^2}{12}$ et que $|Z| > \frac{\Delta_2}{2} \Rightarrow D_2 = \Delta_2^2$.\\ - -\begin{align*} -D &= D_1 Pr(|Z| \leq \frac{\Delta_1}{2}) + D_2 Pr(|Z| > \frac{\Delta_1}{2})\\ -&= \frac{\Delta_2^2}{12} \int_{-\frac{\Delta_1}{2}}^{\frac{\Delta_1}{2}} \frac{1}{\sqrt{2\pi \sigma_Z^2}} exp(-\frac{z^2}{2\sigma_Z^2})dz + \Delta_1^2 \sqrt{\frac{2}{\pi \sigma_Z^2}} \int_{\frac{\Delta_1}{2}}^{+\infty} exp(-\frac{z^2}{2\sigma_Z^2})dz\\ -&= \sqrt{\frac{2}{pi \sigma_z^2}} \left(\frac{\Delta_2^2}{12} \int_{0}^{\frac{\Delta_1}{2}} exp(-\frac{z^2}{2\sigma_Z^2})dz + \Delta_1^2 \int_{\frac{\Delta_1}{2}}^{+\infty} exp(-\frac{z^2}{2\sigma_Z^2})dz\right) -\end{align*} - -\item R est fixé donc M est fixé, et $\Delta_2 = \frac{\Delta_1}{M}$. - -On calcul $\frac{\partial D}{\partial \Delta_1}$ et par magie on trouve 0. - -\end{enumerate} -\end{document} \ No newline at end of file diff --git a/455-Codage_Sources/TD/TD6/1.png b/455-Codage_Sources/TD/TD6/1.png deleted file mode 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\begin{align*} -y_2 = (0100) \Rightarrow \hat{x_2} = 000\text{ }111\text{ }000\text{ }000 -\end{align*} - -\item On a $P(X=0) = 0.8$. Et $R = \frac{j}{Mj} = \frac{1}{M} $ bits/symbole.\\ -Pour M=2, y=0 si $n<\frac{M}{2}$ et 0 sinon, d'où:\\ -\begin{tabular}{|c|c|c|c|c|c|} -\hline -$\underline{X}$ & p & n & y & $\hat{\underline{x}}$ & d \\ -\hline -\hline -00 & $0.8^2$ & 0 & 0 & 00 & 0 \\ -\hline -01 & $0.8\times 0.2$ & 1 & 1 & 11 & 1 \\ -\hline -10 & $0.2 \times 0.8$ & 1 & 1 & 11 & 1 \\ -\hline -11 & $0.2^2$ & 2 & 1 & 11 & 0 \\ -\hline -\end{tabular} - -\begin{align*} -d(x,\hat{x}) &= \sum x_i \oplus \hat{x_i}\\ -D_2 &= \frac{1}{2}(0*0.8^2+1*0.8*0.2+1*0.8*0.2+0)\\ -\end{align*} -Pour $M=1$ on a $D_1 = 0$ et $R_1 = 1$ bits/symbole.\\ -Et l'autre qui demande à aller beaucoup plus vite... - - -n=0,...,M -$p_n = C_M^n*0.2^n * 0.8^{m-n}$ -Si $n<\frac{M}{2}$ $\hat{x}$ contient M 0 et x N 1, donc $d_n = n$ -Si $n\geq \frac{M}{2} $ $\hat{x}$ contient M bits de 1 et X M-n bits de 0 donc $d_n = M-n$ - -$D_n = \sum_{n=0}^{\lceil\frac{M}{2}\rceil - 1} C_M^n 0.2^n 0.8^{M-n} + \sum_{n = \lceil\frac{M}{2}}^{M} C_M^n 0.2^n 0.8^{M-N} (M-n) = \frac{1}{M}$ - -\item -\begin{align*} -x \longrightarrow y \longrightarrow Codeur \\ -1 \longrightarrow \frac{1}{M} \longrightarrow \frac{1}{M} H(Y) = R\\ -P(Y=0) = \sum_{n=0}^{\lceil\frac{M}{2}\rceil - 1} C_M^n 0.8^{M-n} 0.2^n\\ -P(Y=1) = 1-P(Y=0)\\ -H(Y) = -P(Y=0)log_2(P(Y=0)) - P(Y=1) log_2(P(Y=1)) -\end{align*} -\end{enumerate} - - -\section*{Exercice I-TD4: Codage prédictif d'une image} - -\begin{align*} -\underline{\underline{X}} = [\underline{\underline{X}}(:,1),\underline{\underline{X}}(:,2),...,\underline{\underline{X}}(:,256)]\\ -\underline{X_1} = \begin{pmatrix} -\underline{\underline{X}}(:,1)\\ -\underline{\underline{X}}(:,2)\\ -\vdots\\ -\underline{\underline{X}}(:,256) -\end{pmatrix} -\end{align*} - -\begin{enumerate} -\item L'hypothèse gaussienne est fausse, mais permet d'obtenir de bon ordre de grandeurs, $D(R) \leq D_g(R)$\\ - -\item On a $D(R) = \sigma_x^2 * 2^{-2R}$ pour une source gaussienne.\\ -$RSAB_{db} = 10 log_10 \frac{\sigma_x^2}{\sigma_x^2 2^{-2R}} \approx 6R$\\ -$RSAB_{db} \geq 30 dB \Rightarrow R \geq 5$ bits/pixel. - -\item \begin{align*} -\hat{x}(n) = [a_1 -r = E(x(n)x_p(n) = \begin{pmatrix} -\mu_x(1)\\ -\vdots -\mu_x(p) -\end{pmatrix}\\ -R = \begin{pmatrix} -\mu_x(0) & &\\ -\vdots & \mu_x(|i-j|) & \\ -\mu_x(p) & & -\end{pmatrix} -\end{align*} - -Des résultats random :\\ -Par définition : \\ -$\sigma_e^2 = (e^2(n))= E((x(n)-\hat{x(n)})^2)$ -Par minimisation de $\sigma_e^2$ donc en dérivant: -$a_p = R^{-1} r$ puisque R est inversible.\\ -\begin{align*} -\sigma_e^2 = \mu_x(0) - 2(R^{-1}r)^Tr + (R^{-1}r)^TR(R^{-1}r)\\ -=\mu_x(0) - r^TR^{-1}r -\end{align*} - -\item L'écart entre deux pics est 256. Les ressemblances des pixels d'une même ligne qui sont séparés de 256 pixels dans $\hat{x}$. Les maximums décroissent. - -\item Pour p=1 r vaut $\mu_x(1)$ et $R = \mu_x(0)$, ainsi $a_p = 0.95$. -Et $\sigma_e^2 = 255.95$. - -\item $D_e = \sigma_e^2 2^{-2R}$ -$RSBA_{db} = 10 log_{10} \frac{\sigma_x^2}{D_e} \approx10.4 + 6R = 30.$ -Donc R = 33 bits/symbole - -\item Gain de codage: -$RSAB^{(p)} - RSAB = 10.4 db$ - - - -\end{enumerate} - - - - -\end{document} \ No newline at end of file