{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "2715b8db-a9b9-47be-8f59-340cb0660279",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "| VEC1 : BaseS temporelleS | => [VEC2 BaseS FréquentielleS](VEC2_bases_frequentielles.ipynb) |\n",
    "|---------------------------|------------------|\n",
    "\n",
    "---\n",
    "# VEC1 - BaseS temporelleS\n",
    "---\n",
    " - [A](#A--Espace-vectoriel-des-fonctions) :  Espaces vectoriels de fonctions\n",
    " - [B](#Espace-vectoriel-des-suites-r%C3%A9elles) : E.v. des suites réelles\n",
    " - [C](#C--Prolongement-aux-suites-r%C3%A9elles-%C3%A0-support-born%C3%A9) : Prolongement aux suites réelles bornées\n",
    "   * [C1](#C1---Isomorphisme-(bijection)-avec-un-sous-espace-des-suites) : Bijection avec un sous espace\n",
    "   * [C2](#C2---Base-temporelle-orthogonale) : Base temporelle orthogonale\n",
    "   * [C3](#C3---Projection-(%C3%A9chantillonnage)-de-la-base-RZ\\mathbb{R}^\\mathbb{Z}-dans-Rb3\\mathbb{R}_3^b) : Projection (échantillonnage) dans les suites bornées\n",
    " - [D](#D---Prolongement-aux-suites-N-p%C3%A9riodiques) : Prolongement aux suites N-périodiques\n",
    " - [E](#E---Prolongement-aux-suites-paires-N-p%C3%A9riodiques) : Prolongement aux suites paires N-périodiques\n",
    "---\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "Prenons un ensemble de 3 coordonnées (1, 2, 3) et plongeons-les dans un espace vectoriel de base $W$ avec :  \n",
    "\n",
    "$\\vec{v} = 1.\\vec{w_0} + 2.\\vec{w_1} + 3.\\vec{w_3} = (1,2,3)_W$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "71f6feed-3cba-43ce-8094-52dd2614efa2",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "v =\n",
      "\n",
      "   1\n",
      "   2\n",
      "   3\n",
      "\n"
     ]
    }
   ],
   "source": [
    "clear all;\n",
    "close all;\n",
    "clc;\n",
    "\n",
    "v = [1; 2; 3]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7566e4fb-fc03-40c7-bb3b-25528f5baadc",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "Elles peuvent représenter différents objets mathématiques d'un espace vectoriel :\n",
    "\n",
    "- Des matrices 2x2 symétriques\n",
    "- un polynôme\n",
    "- un vecteur de l'espace euclidien\n",
    "- une couleur\n",
    "- un signal continu\n",
    "- un signal discret\n",
    "- ...\n",
    "\n",
    "Le tout est de chosir une base $(\\vec{w_0}, \\vec{w_1}, \\vec{w_2})$ et de dire :\n",
    "\n",
    "$\\vec{v} = 1.\\vec{w_0} + 2.\\vec{w_1} + 3.\\vec{w_3}$\n",
    "\n",
    "Ainsi pour un espace des matrices 2x2 symétriques"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "123c860e-140d-4b4b-85c7-38048ff0f6f5",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vectv =\n",
      "\n",
      "   1   3\n",
      "   3   2\n",
      "\n"
     ]
    }
   ],
   "source": [
    "w0=[1 0;\n",
    "    0 0];\n",
    "w1=[0 0;\n",
    "    0 1];\n",
    "w2=[0 1;\n",
    "    1 0];\n",
    "v;\n",
    "Vectv = v(1)*w0 +v(2)*w1 +v(3)*w2 \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ac25226-cee2-4207-95ff-f03cc67107d6",
   "metadata": {
    "tags": [],
    "toc-hr-collapsed": true
   },
   "source": [
    "# A- Espace vectoriel des fonctions \n",
    "---\n",
    "Si nous voulons représenter un signal continu à support infini, il faut choisir une base avec des vecteurs de ce type.  \n",
    "Comme la base se veut temporelle : il faudrait choisir des fonctions localisées (impulsions $p(t)$) autour d'un instant mais qui s'étalent à l'infini.\n",
    "\n",
    "L'idée est d'utiliser des impulsions temporelles $t\\mapsto p(t)$ :\n",
    ">  $(1,2,3)_W = \\vec{v} = 1.\\vec{w_0} + 2.\\vec{w_1} + 3.\\vec{w_3} \\quad \\leftrightarrow \\quad  f: t\\mapsto 1.p(t) + 2.p(t-1) + 3.p(t-2)\\quad $  \n",
    ">  avec $f(0)=1, f(1) = 2, f(2) = 3$ par exemple \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dbdba807-f955-4739-94ac-10ab708b27ff",
   "metadata": {
    "tags": []
   },
   "source": [
    "On peut penser aux fonctions portes, mais aussi aux gaussiennes, sinus cardinal, etc.\n",
    "\n",
    "Ainsi avec des sinus cardinaux et en décidant arbitrairement que la composante 0 est autour de l'instant 0, la composante 1 autour de l'instant 1, etc.\n",
    "\n",
    "Cela donnerait :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "e238d7a3-ae0f-4ec8-83bf-8228f70b7abb",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t=-5:0.01:5;    % vecteur temps presque continus ;-)\n",
    "\n",
    "w0 = sinc(t);\n",
    "w1 = sinc(t-1);\n",
    "w2 = sinc(t-2);\n",
    "\n",
    "Vect_v_sinc = v(1)*w0 +v(2)*w1 +v(3)*w2 ;\n",
    "\n",
    "plot(t,Vect_v_sinc,t,w0,t,w1,t,w2); hold on; grid on;\n",
    "plot(0:2,v,'dr')\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99984391-00b7-4144-abda-da42131eef28",
   "metadata": {},
   "source": [
    "On voit bien que la dimension 3 de cette base (les vecteurs sont libres) ne représente qu'un sous-ensemble très restreint de l'espace des fonctions qui est de dimension infinie indénombrable.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77dd8d38-b79c-4d0e-930e-f160b4928159",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Espace vectoriel des suites réelles\n",
    "---\n",
    "\n",
    "Cherchons donc parmis les signaux discrets (échantillonnés), mathématiquement définis comme des suites numériques.\n",
    "\n",
    "Prenons 3 vecteurs libres de l'espace des suites entières infinies. Par exemple en discrétisant arbitrairement les sinus cardinaux précédents.\n",
    ">  $(1,2,3)_W = \\vec{v} = 1.\\vec{w_0} + 2.\\vec{w_1} + 3.\\vec{w_3}  \\quad \\leftrightarrow \\quad  f: k\\mapsto 1.p[k] + 2.p[k-1] + 3.p[k-2]\\quad$  \n",
    ">  avec la suite $(\\dots, 1, 2, 3, \\dots)$ par exemple \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0de2bf35-0c41-4b39-8e25-9b800aa8018d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "k=-10:1:10;    % vecteur temps discret presque infinis ;-)\n",
    "\n",
    "w0 = sinc(k);\n",
    "w1 = sinc(k-1);\n",
    "w2 = sinc(k-2);\n",
    "\n",
    "Vectv = v(1)*w0 +v(2)*w1 +v(3)*w2 ;\n",
    "\n",
    "\n",
    "stem(k,Vectv,'k');hold on;\n",
    "plot(t,Vect_v_sinc,'k--')\n",
    "\n",
    "\n",
    "stem(k,w0,'r')\n",
    "stem(k,w1,'g')\n",
    "stem(k,w2,'b'); \n",
    "plot(t,sinc(t-2),'b--')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7361315b-0bb6-43ce-a69d-d8e51b377773",
   "metadata": {
    "tags": []
   },
   "source": [
    "Et oui ! En échantillonant à la période des sinus cardinaux on obtient un vecteur de base temporel qui est l'impulsion unité (qui est au discret ce que l'impulsion de Dirac est au continu)\n",
    "\n",
    "> Définissons la **fonction impulsion unité**  \n",
    ">$\\delta_0 : \\mathbb{Z} \\to \\mathbb{R}$ parfois $\\mathbb{C}$  \n",
    ">$\\quad k \\mapsto \\begin{cases} 1 & \\text{ si } k = 0 \\\\ 0 & \\text{ sinon}\\end{cases}$  \n",
    ">  \n",
    "> Et l'impulsion centrée en $a$ :\n",
    "> $\\delta_a : k \\mapsto \\delta_0[k-a]=\\begin{cases} 1 & \\text{ si } k = a \\\\ 0 & \\text{ sinon}\\end{cases}$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46ff813d-408c-49f9-9274-ae7fc2809657",
   "metadata": {
    "tags": []
   },
   "source": [
    "# C- Prolongement aux suites réelles à support borné\n",
    "---\n",
    "\n",
    "Le prolongement borné consiste à utiliser des impulsions à durée finie (nulles pour $|n|>N_0$). Le prolongement est alors :\n",
    "\n",
    ">  $(1,2,3)_W = \\vec{v} = 1.\\vec{w_0} + 2.\\vec{w_1} + 3.\\vec{w_3}  \\quad \\leftrightarrow \\quad  f: t\\mapsto 1.p[k] + 2.p[k-1] + 3.p[k-2]\\quad \\;$\n",
    "> tel que $\\;k\\notin[\\![0; N [\\![ \\implies f[k]=0$   \n",
    ">  avec la suite $(\\dots, 0, 1, 2, 3, 0, \\dots)$ par exemple \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c82fbe24-d5f9-4e7d-aea1-854c71f13983",
   "metadata": {},
   "source": [
    "Les suites réelles infinies ne sont pas manipulable dans le cas pratique, nous allons considérer une fenêtre d'observation.\n",
    "> Soit la **fonction porte** ou **fenêtre naturelle** de largeur N :  \n",
    "> $\\Pi_N : k \\mapsto  \\left\\{ \\;\\begin{array}{ll} 1 & \\text{ si } 0 \\leq k < N \\\\ 0 & \\text{sinon}\\end{array}\\right.$\n",
    "\n",
    "\n",
    "On peut ainsi définir l'espace des suites réelles et différents sous espaces vectoriels à support fini de N échantillons :\n",
    "> **Suite réelle à support borné de N échantillons :**  \n",
    "> $\\mathbb{R}^N_b = \\mathbb{R}^{[\\![0\\,;N [\\![} = \\left(u_k . \\Pi_N[k]\\right)_{k\\in\\mathbb{Z}}$  \n",
    "> Autrement dit \"l'ensemble des suites réelles multipliées par une fenêtre d'observation de N échantillons\"\n",
    "\n",
    "\n",
    "On obtient ainsi des sous-espaces vectoriels (s.e.v.) de $\\mathbb{R}^{\\mathbb{Z}}$ que nous notons donc :\n",
    ">  $(u_k)_{k\\in\\mathbb{Z}}\\quad$ noté $\\mathbb{R}^{\\mathbb{Z}}\\quad$ : l'**espace des suites réelles**  \n",
    ">  $(u_k)_{k\\in\\mathbb{N}}\\quad$ noté $\\mathbb{R}^{\\mathbb{N}}\\quad\\subset\\mathbb{R}^{\\mathbb{Z}}$ : l'**espace des suites réelles causales** (nulles pour $k<0$)   \n",
    ">  $\\left(u_k . \\Pi_N[k]\\right)_{k\\in\\mathbb{Z}}$ noté $\\mathbb{R}^{[\\![0 \\:;\\; N [\\![}$ parfois $\\mathbb{R}^N_B \\subset\\mathbb{R}^{\\mathbb{N}}\\subset\\mathbb{R}^{\\mathbb{Z}}$ : l'**espace des suites réelles à support finit de longueur N** \n",
    "\n",
    "Pour obtenir une base de $\\mathbb{R}^3_B$ on utilise **les impulsions unités**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "be53943e-d128-4721-b56c-b6d0ebee205e",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "% La fonction delta est VECTORISEE : \n",
    "%    k peut être une matrice de valeurs\n",
    "function [v] = delta_0(k)\n",
    "    [Nrang,Ncol] = size(k);   \n",
    "    v = zeros(Nrang, Ncol);\n",
    "    for rang_k = 1:Nrang\n",
    "        for col_k = 1:Ncol\n",
    "                if k(rang_k,col_k) == 0\n",
    "                    v(rang_k,col_k)=1;\n",
    "                end                 \n",
    "        end % boucle sur colonne de k\n",
    "    end % boucle sur rangée de k\n",
    "end \n",
    "\n",
    "w0 = delta_0(k);    % w0 matrice rangée (1,21)\n",
    "w1 = delta_0(k-1);  % w1,w2  idem mais\n",
    "w2 = delta_0(k-2);  %  avec le 1 pour k-1 = 0 etc. \n",
    "\n",
    "Vectv = v(1)*w0 +v(2)*w1 +v(3)*w2 ; % Somme de matrices (1,21)\n",
    "\n",
    "subplot(4,1,1); stem(k,v(1)*w0,'r') ; axis([-5,5,0,3]) ; set(gca, \"ytick\", 0:4); grid on; %1.w0 composante rouge\n",
    "subplot(4,1,2); stem(k,v(2)*w1,'g') ; axis([-5,5,0,3]) ; set(gca, \"ytick\", 0:4); grid on;%2.w1 composante verte\n",
    "subplot(4,1,3); stem(k,v(3)*w2,'b') ; axis([-5,5,0,3]) ; set(gca, \"ytick\", 0:4); grid on;    %3.w2 composante bleue\n",
    "subplot(4,1,4); \n",
    "stem(k,Vectv,'k'); ; axis([-5,5,0,3]) ; set(gca, \"ytick\", 0:4); grid on; % somme des composantes noire"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7153796c-3bad-4f29-81d5-6b61250b78f3",
   "metadata": {
    "tags": []
   },
   "source": [
    "## C1 - Isomorphisme (bijection) avec un sous espace des suites\n",
    "\n",
    "Remarquez que l'on vient de prolonger le vecteur $\\vec{v}\\in\\mathbb{R}^3$ avec des $0$ pour se plonger dans un sous espace vectoriel des suites réelles $(u_n)_{n\\in\\mathbb{Z}}$ noté $\\mathbb{R}^{\\mathbb{Z}}$.\n",
    "\n",
    "\n",
    "\n",
    "On peut faire une bijection entre un vecteur de $\\vec{v}\\in\\mathbb{R}^3$ (de dimention 3) avec une suite observable de $\\mathbb{R}^3_b$ qui est un s.e.v. de dimension 3 de l'espace des suites réels.\n",
    "\n",
    "$\\vec{v}=(1,2,3)_{B_3}=1.e_1 + 2.e_2 + 3.e_3 = \\begin{array}{lll} &1. &(1,0,0)\\\\ +& 2.&(0,1,0)\\\\ + &3.&(0,0,1) \\end{array}= \\left[\\begin{array}{c}1 \\\\2\\\\3 \\end{array}\\right]_{B_3} \\leftrightarrow  \\left[\\begin{array}{c}1 \\\\2\\\\3 \\end{array}\\right]_{B_3^b} = \\begin{array}{lll} & 1. &(\\dots, 1, 0,0,\\dots)\\\\ +& 2.&(\\dots, 0, 1,0,\\dots) \\\\ +& 3.&(\\dots, 0, 0, 1,\\dots)\\end{array}=1.\\delta_0 + 2.\\delta_1 + 3.\\delta_2$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3410b4a1-f7d5-477f-aa53-80fa62022929",
   "metadata": {},
   "source": [
    "Les espaces étant isomorphiques, on associera et notera indiféremment la base temporelle $B_3^b$ et la base euclidienne $B_3$.\n",
    "\n",
    "\n",
    "Nous définissons donc les bases canoniques de ces espaces :\n",
    "\n",
    "\n",
    ">  $B_{\\mathbb{Z}} = \\left( \\delta_k \\right)_{k\\in\\mathbb{Z}}\\quad$ avec $\\mathbb{R}^{\\mathbb{Z}} = \\mathop{vect}\\left(B_{\\mathbb{Z}}\\right)$  \n",
    ">  $B_{\\mathbb{N}} = \\left( \\delta_k \\right)_{k\\in\\mathbb{N}}\\quad$ avec $\\mathbb{R}^{\\mathbb{N}} = \\mathop{vect}\\left(B_{\\mathbb{N}}\\right)$  \n",
    ">  $B_N^b = \\left( \\delta_k \\right)_{k\\in[\\![0 \\:;\\; N [\\![}\\quad$ avec $\\mathbb{R}^N_b = \\mathop{vect}\\left(B_N^b\\right)$ où le \"b\" signifie \"support Borné\" \n",
    "\n",
    "\n",
    "Numériquement, nous pouvons définir les matrices de passsages entre ces bases en prenant pour $\\mathbb{R}^{\\mathbb{Z}}$ un nombre de point plus grand que 3 : \n",
    "\n",
    "**Inutile et impossible de représenter un signal sur un support infini !**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "52111c6c-610b-4c58-848e-b4f5af00ed8a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M_Bz_B3b =\n",
      "\n",
      "  0  0  0\n",
      "  1  0  0\n",
      "  0  1  0\n",
      "  0  0  1\n",
      "  0  0  0\n",
      "  0  0  0\n",
      "  0  0  0\n",
      "  0  0  0\n",
      "  0  0  0\n",
      "\n",
      "v_B3b =\n",
      "\n",
      "   0\n",
      "   1\n",
      "   2\n",
      "   3\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "\n"
     ]
    }
   ],
   "source": [
    "function v = delta(t)\n",
    "    v = (t==0); % vrai = 1, faux = 0\n",
    "end \n",
    "%% t == 0 se fait pour chacun des éléments de t !\n",
    "%% t peut être une matrice : c'est la VECTORISATION\n",
    "\n",
    "%% On prend un support de z infini ou presque (R^8)\n",
    "k= transpose(-1:7); % temps de Bz en colonnes\n",
    "\n",
    "%% On donne les coordonnées de w0 à w2 dans la base\n",
    "w0 = delta(k);    % maintenant en colonnes\n",
    "w1 = delta(k-1); \n",
    "w2 = delta(k-2); \n",
    "\n",
    "%% La matrice de passage consiste à lister en colonne\n",
    "% les vecteur de la base delta_n\n",
    "M_Bz_B3b = [w0, w1, w2] % Matrice de colonnes\n",
    "\n",
    "%%Ainsi notre vecteur dans la base Bz s'obtient en\n",
    "%% multipliant les 3 coordonnées par la matrice de passage\n",
    "v_B3b = M_Bz_B3b * v"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b35e3409-81d8-4bc8-aee3-7586080860b6",
   "metadata": {
    "tags": []
   },
   "source": [
    "## C2 - Base temporelle orthogonale\n",
    "\n",
    "Nous pouvons vérifier que cette base ($w_0$, $w_1$, $w_2$) de suite réelles est bien orthogonale et normée. Pour cela il faut effecteur tout les produits scalaires et vérifier :\n",
    "\n",
    "- $<w_k, w_l> = 0$ si $k\\neq l$\n",
    "- $<w_k, w_k> = \\|w_k\\|^2 = 1 $\n",
    "\n",
    "Remarquons que nous n'avons pas défini le produit scalaire entre suites numériques, mais comme nous avons une base $B_N^b$ on sait que l'on peut l'obtenir en multipliant à gauche un vecteur par sa transposée : \n",
    "\n",
    "$ <w_k, w_l> = {}^T\\!w_k|_{B_N^b} \\,.\\, w_l|_{B_N^b}$\n",
    "\n",
    "En prenant la matrice des vecteurs colonnes, qui est la matrice de passage $M = M_{B_3^b\\leftarrow B_3} = \\left[ w_k|_{B_3^b}\\right]_{0\\leq k < 3}$, et en la transposant : on obtient la matrice de tous les vecteurs transposés ${}^T\\!M = \\left[{}^T\\!w_k|_{B_3^b}\\right]_{0\\leq k < 3}$. On peut donc calculer tous les produits scalaires en multipliant ces deux matrices :\n",
    "\n",
    "${}^T\\!M.M = \\left( <w_j,w_k>\\right)_{(j,k)\\in[\\![0 ; 3[\\![^2} = I_d$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "80ffddc2-ee0a-4c7e-8b39-344cc55f8b7d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M_Bz_B3b =\n",
      "\n",
      "  0  0  0\n",
      "  1  0  0\n",
      "  0  1  0\n",
      "  0  0  1\n",
      "  0  0  0\n",
      "  0  0  0\n",
      "  0  0  0\n",
      "  0  0  0\n",
      "  0  0  0\n",
      "\n",
      "ans =\n",
      "\n",
      "  0  1  0  0  0  0  0  0  0\n",
      "  0  0  1  0  0  0  0  0  0\n",
      "  0  0  0  1  0  0  0  0  0\n",
      "\n",
      "ans =\n",
      "\n",
      "   1   0   0\n",
      "   0   1   0\n",
      "   0   0   1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%% Fonctions anonymes ou fonctions \"lambda\"\n",
    "% @(t) (t==0)   \n",
    "% est une fonction d'un parametre t \"déclarée à la volée\"\n",
    "delta = @(t) (t==0);  % on appelle fa fonction anonyme \"delta\"\n",
    "\n",
    "%% Broadcast de variable (pas de français connus)\n",
    "% Si une dimention manque à une variable, elle est \n",
    "% ajoutée en complétant \"intelligemment\"...\n",
    "k = transpose(-1:7); % matrice colonne (8,1)\n",
    "j = 0:2;  % matrice ligne (1,3)\n",
    "M = k-j;\n",
    "% l'expression k-j de type (8,1)-(1,3) non définie en maths\n",
    "% alors \n",
    "% la colonne k (8,1) est recopiée sur 3 colonnes => (8,3)\n",
    "% la rangée  j (1,3) est recopiée sur 8 rangées  => (8,3)\n",
    "% la soustraction matricielle (8,3) peut se faire par BROADCAST\n",
    "\n",
    "% par fonction anonyme vectorisée du broadcast de k-j \n",
    "M_Bz_B3b = delta(k-j) \n",
    "transpose(M_Bz_B3b)\n",
    "transpose(M_Bz_B3b) * M_Bz_B3b\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48d05343-2347-4668-9f7c-9b1048724745",
   "metadata": {},
   "source": [
    "La base est bien orthonormée et donc les matrices de passages sont orthogonales :\n",
    "* orthogonaux : les éléments hors diagonale ($j\\neq k$) sont les produits scalaires et sont nuls\n",
    "* normés : les éléments diagonaux ($j=k$) sont la norme de chaque vecteur et vallent 1 \n",
    "\n",
    "Les matrices de passage ne sont pas carrées car la dimension des espaces est différente (l'un est de dimension infinie dénombrable et l'autre de dimension 3)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b46e2be6-79c7-4c9e-9580-6b33de48e583",
   "metadata": {
    "tags": []
   },
   "source": [
    "## C3 - Projection (échantillonnage) de la base $\\mathbb{R}^\\mathbb{Z}$ dans $\\mathbb{R}_3^b$\n",
    "\n",
    "Si l'on prend un vecteur $\\vec{f}\\in\\mathbb{R}^\\mathbb{Z}$ (donc une suite réelle à support infini), on peut la projeter (ou décomposer dans l'espace des suites de 3 points : $\\mathbb{R}_3^b$).\n",
    "\n",
    "\n",
    "Comme la base est une base orthonormée (b.o.n.) la projection est simplement :\n",
    "\n",
    "$\\mathop{proj}_{/B_3^b}(\\vec{f}) = \\left[\\begin{array}{l} <\\vec{f},\\vec{w_0}> \\\\ <\\vec{f},\\vec{w_1}>\\\\<\\vec{f},\\vec{w_2}>\\end{array}\\right]_{B_3^b} = \\left[\\begin{array}{l} <\\vec{w_0},\\vec{f}> \\\\ <\\vec{w_1},\\vec{f}>\\\\<\\vec{w_2},\\vec{f}>\\end{array}\\right]_{B_3^b} = \\left[\\begin{array}{l} {}^T\\!{w_0|}_{B_Z}.g|_{B_Z}\\\\ {}^T\\!{w_1|}_{B_Z}.g|_{B_Z}\\\\{}^T\\!{w_2|}_{B_Z}.g|_{B_Z}\\end{array}\\right]_{B_3^b}= \\underbrace{\\left[\\begin{array}{l} {}^T\\!{w_0|}_{B_Z} \\\\ {}^T\\!{w_1|}_{B_Z}\\\\ ^T\\!{w_2|}_{B_Z}\\end{array}\\right]}_{M_{B_3^b\\leftarrow B_Z}}.g|_{B_Z}=\\underbrace{{\\begin{array}{l} { }\\\\ {}  \\\\ { } \\end{array}}^T\\!\\left[\\begin{array}{l} {w_0|}_{B_Z} \\\\ {w_1|}_{B_Z}\\\\ {w_2|}_{B_Z}\\end{array}\\right]}_{{}^T\\!M_{B_Z\\leftarrow B_3^b}}.g|_{B_Z} $\n",
    "\n",
    "En utilisant les coordonnées pour calculer les produits scalaires matriciellement on obtient :\n",
    "\n",
    "$\\mathop{proj}_{/B_3^b}(\\vec{f})|_{B_3^b} = M_{B_3^b\\leftarrow B_Z} . \\vec{f}|_{B_Z} = {}^T\\!M_{B_Z\\leftarrow B_3^b} . \\vec{f}|_{B_Z}$\n",
    "\n",
    "\n",
    "On en déduit ainsi les produits scalaires que l'on peut associer aux deux espaces $\\mathbb{R}^\\mathbb{Z}$ dans $\\mathbb{R}^3_b$ :\n",
    "\n",
    "* Dans $\\mathbb{R}^\\mathbb{Z}$ on note le produit scalaire $<\\!< f, g>\\!> = \\sum\\limits_{k\\in\\mathbb{Z}}f[k].g[k]$\n",
    "* Dans $\\mathbb{R}^N_b$ on note le produit scalaire $<\\!< f, g>\\!>_p = \\sum\\limits_{0 \\leq k < N}f[k].g[k]$\n",
    "\n",
    "Où l'on note $f[k]$ le terme de rang $k$ d'une suite $(f_k)=(f[k])$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "bf6756c9-56d4-4675-a775-79dfd8a01aff",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "norme_erreur =  70.271\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%% prenont la fonction t^2+1 échantillonnée dans Bz\n",
    "f_Bz = k.^2 + 1;\n",
    "\n",
    "%% Projetons sur B3b: on note p_f la projectio nde f\n",
    "p_f_B3b = transpose(M_Bz_B3b) * f_Bz;\n",
    "\n",
    "%% Revenons dans la base de départ (recomposition)\n",
    "p_f_Bz = M_Bz_B3b * p_f_B3b ; \n",
    "\n",
    "%% On regarde la norme de l'erreur \n",
    "%% pour voir s'il manque du signal \n",
    "\n",
    "err_pf_Bz = (f_Bz - p_f_Bz);\n",
    "norme_erreur = sqrt(transpose(err_pf_Bz)*err_pf_Bz)\n",
    "\n",
    "%% Affichons ces signaux\n",
    "stem(k,f_Bz,'k--+');hold on;\n",
    "stem(k,p_f_Bz,'r')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16131f00-769c-40d8-90ed-b1a68f90ee7b",
   "metadata": {},
   "source": [
    "On voit bien que $\\vec{f}$ est différent de sa projection $\\mathop{proj}_{B_3^b}(\\vec{f})$ sur l'espace des suites de 3 points.\n",
    "\n",
    "\n",
    "On verra que l'on peut trouver une b.o.n de l'espace des suite de 3 points qui soit basée sur les fréquences et non plus sur le temps.\n",
    "\n",
    "> changer de base dans $\\mathbb{R}^N_b$ avec une base fréquencielle sera  \n",
    ">  effectuer une TFSD (Transformée de Fourier des Signaux Discrets)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bca58fcf-fb3b-418b-8cd5-8b5b506166db",
   "metadata": {
    "tags": []
   },
   "source": [
    "# D - Prolongement aux suites N-périodiques\n",
    "---\n",
    "\n",
    "Le prolongement périodique consiste à utiliser des impulsions $p$ qui soient N-périodiques.\n",
    "\n",
    "Le prolongement est alors :\n",
    ">  $(1,2,3)_W = \\vec{v} = 1.\\vec{w_0} + 2.\\vec{w_1} + 3.\\vec{w_3}  \\quad \\leftrightarrow \\quad  f: t\\mapsto 1.p[k] + 2.p[k-1] + 3.p[k-2]$\n",
    "> tel que $\\;\\forall k, f[k]=f[k+N]$   \n",
    ">  avec la suite $(\\dots, 1, 2, 3,\\, 1, 2, 3,\\, 1, 2, 3,\\dots)$ par exemple \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "facca253-c686-45df-9490-28216eadde1d",
   "metadata": {},
   "source": [
    "\n",
    "Dans ce cas, il faut prendre des vecteurs de la base qui soient de période 3 et localisés dans le temps. On construit alors simplement une version périodique des vecteurs de $\\delta_j$. \n",
    "\n",
    "> On définit alors un signal appelé **peigne d'impulsions** de période N  \n",
    "> noté $I\\!I\\!I_N$ (letrre cyrilique Sha) défini par :  \n",
    "> $I\\!I\\!I_N := \\sum\\limits_{j\\in\\mathbb{Z}}  \\delta_{jN} : k \\mapsto \\delta_0[k\\mod{}N]\\quad$ \n",
    "\n",
    "On définit donc \n",
    "> le s.e.v. des **suites réelles de période N** et sa base comme :  \n",
    ">  $(u_k = u[k \\mod N ])_{k\\in\\mathbb{Z}}\\quad$ noté $\\mathbb{R}^{[\\![0 \\:;\\; N [\\![_P}\\quad\\notin\\mathbb{R}^{\\mathbb{N}}\\quad\\subset\\mathbb{R}^{\\mathbb{Z}}$ : l'**espace des suites réelles de période N**  \n",
    "> $B_N^p = \\left(k\\mapsto I\\!I\\!I_N[k - j]\\right)_{j\\in\\mathbb{Z}}$ et donc $\\mathbb{R}^{[\\![0 \\:;\\; N [\\![_P} = \\mathop{vec}(B_N^p)$\n",
    "\n",
    "Numériquement traçons les vecteurs de $B_3^P$ dans une base relativement proche de $B_Z$\n",
    "\n",
    "Ainsi $\\mathbb{R}^3$ est isomorphe à $\\mathbb{R}^3_p$ qui est un s.e.v de $\\mathbb{R}^{\\mathbb{Z}}$.\n",
    "Donc pour le vecteur v :\n",
    "\n",
    "$ v \\in \\mathbb{R}^3 \\leftrightarrow  v|_{B_3^p} \\in \\mathbb{R}^3_b $\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "d9645812-a4a3-48b2-8481-928f01f62bf6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%% Fonction anonyme \"peigne\" Vectorisée à 2 variables\n",
    "peigne = @(k, N) delta(mod(k,N));\n",
    "% Si k est matrice mais N vecteur ou scalaire : \n",
    "% il y aura broadcast de N\n",
    "\n",
    "% Broadcast de k-j en matrice (k,j)\n",
    "M_Bz_B3p = peigne(k-j,3);\n",
    "%  _______________________\\ j\n",
    "% |                       /\n",
    "% |  w0_Bz  w1_Bz  w2_Bz\n",
    "% |\n",
    "%\\|/\n",
    "% k\n",
    "\n",
    "w0_Bz = M_Bz_B3p(:,1);\n",
    "w1_Bz = M_Bz_B3p(:,2);\n",
    "w2_Bz = M_Bz_B3p(:,3);\n",
    "\n",
    "couleurs=['r';'g';'b'];\n",
    "for id_j=1:3\n",
    "    subplot(5,1,id_j)\n",
    "    stem(k,M_Bz_B3p(:,id_j),couleurs(id_j));\n",
    "end\n",
    "\n",
    "v_Bz = M_Bz_B3p * v ;\n",
    "\n",
    "subplot(5,1,[4 5])\n",
    "stem(k,v_Bz,'k--')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51bb63b3-4e76-4a8c-8105-624a379fcbb2",
   "metadata": {},
   "source": [
    "Ce prolongement périodique est pratique si l'on utilise une base fréquentielle, car comme pour les séries de Fourier si le signal est sur un support périodique de période $T_0$, alors seules les fréquences multiples de $F_0$ seront utiles. On \n",
    "obtiendra ainsi : \n",
    "\n",
    "> la Transformée de Fourier Discrète (TFD, où FFT est un algorithme de TFD efficace)  \n",
    "> est le passage dans une base fréquentielle (complexe) de la base temporelle $B_3^p$ de $\\mathbb{R}^3_p$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40ce65a8-6403-4fa2-8744-1033201cc477",
   "metadata": {
    "tags": []
   },
   "source": [
    "La base est clairement orthogonale (orthogonalité dans le temps : jamais non nul en même temps) par contre la norme au carré de chaque vecteur vaut le nombre d'impulsions dans le temps. Dans une base de temps infinie, ce signal est de norme infinie, car d'énergie infinie puisque périodique. On devrait plutôt parler de puissance moyenne et sommer uniquement sur une période :\n",
    "\n",
    "\n",
    "> Dans $\\mathbb{R}^{\\mathbb{Z}}$ avec le produit scalaire $<\\!<f,g>\\!>=\\sum_{k\\in\\mathbb{Z}} f[k].g[k]$ donne l'énergie !   \n",
    "> Dans $\\mathbb{R}^N_p$ avec le produit scalaire $<\\!<f,g>\\!>_p=\\sum_{0\\leq k < N} f[k].g[k]$ donne l'énergie sur une période. \n",
    "> Dans $\\mathbb{R}^N_p$ avec le produit scalaire $<\\!<f,g>\\!>_p=\\frac{1}{N}\\sum_{0\\leq k < N} f[k].g[k]$ donne la puissance moyenne.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "46e9b64f-4b33-4eb9-bbf6-c0c65e359941",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N =  3\n",
      "ans =\n",
      "\n",
      "   3   0   0\n",
      "   0   3   0\n",
      "   0   0   3\n",
      "\n",
      "ans =\n",
      "\n",
      "   1   0   0\n",
      "   0   1   0\n",
      "   0   0   1\n",
      "\n",
      "ans =\n",
      "\n",
      "   0.33333   0.00000   0.00000\n",
      "   0.00000   0.33333   0.00000\n",
      "   0.00000   0.00000   0.00000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "scal = @(u,v,N) transpose(v)*u;\n",
    "scalp = @(u,v,N) transpose(v(1:N,:))*u(1:N,:);\n",
    "scalpn = @(u,v,N) transpose(v(1:N,:))*u(1:N,:)/N;\n",
    "\n",
    "N=3\n",
    "scal(M_Bz_B3p,M_Bz_B3p)\n",
    "scalp(M_Bz_B3p,M_Bz_B3p,N)\n",
    "scalpn(M_Bz_B3p,M_Bz_B3b,N)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d57e32c4-0fa7-475e-9b7a-64f84bb549b4",
   "metadata": {},
   "source": [
    "Cette base périodique est donc utile, mais la construction arbitraire de cette période peut ajouter des discontinuités à chaque changement de période (hautes fréquences dans la transformée) et le fait que la fonction ne soit pas paire nécessite de décomposer en cosinus et sinus (a(n), b(n), $n\\geq0$) ou en ondes complexes (c(n), $n\\in\\mathbb{Z}$) avec des fréquences négatives.\n",
    "\n",
    "Sauriez-vous donner un prolongement en suites **paires** périodiques (il faut changer la période) de dimension 3 ?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e573b547-0319-477e-a386-e8e099d82397",
   "metadata": {
    "tags": []
   },
   "source": [
    "# E - Prolongement aux suites paires N-périodiques \n",
    "---\n",
    "\n",
    "Contrairement au continu où il existe un seul prolongement périodique pair possible, en discret il existe plusieurs solutions. \n",
    "\n",
    "Le plus poulaire est de période $2N$ et donne :\n",
    ">  $(1,2,3)_W = \\vec{v} = 1.\\vec{w_0} + 2.\\vec{w_1} + 3.\\vec{w_3}  \\quad \\leftrightarrow \\quad  f: t\\mapsto 1.p[k] + 2.p[k-1] + 3.p[k-2]\\quad \\; | \\;\\forall k, f[k]=f[k+2N]$ et $f[k]=f[-k-1]$  \n",
    ">  avec la suite $(\\dots,\\, 3, 2, 1,\\; 1, 2, 3, 3, 2, 1, \\; 1, 2, 3,\\,\\dots)$ par exemple.   \n",
    ">  **Il y a un déclage temporel !** Car l'indice $k=0$ ne correspond pas au temps 0. On est quasi-pair car $f[k]=f[-k-1]$ et non $f[-k]$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd116710-c501-4cca-82cc-533854893f7c",
   "metadata": {},
   "source": [
    "En continu, le développement en série de cosinus (CT pour Cosine Transform) est facile à définir car :  \n",
    "> Il y n'y a qu'un seul prolongement pair de période double d'un signal à support borné $[0 ; T_0[$. \n",
    "> C'est donc une version réelle des SdF $c(n)$ où l'on prend une période double avec :  \n",
    "> $CT[n] = a(\\frac{n}{2}) = \\int\\limits_0^{T_0} f(t) . \\cos\\left(2\\pi n \\frac{t}{2.T_0}\\right) dt$\n",
    "\n",
    "**Il n'y a pas de $b(n)$ ! Et tout est réel, en revanche la résolution fréquentielle est double : \n",
    "$n\\leftrightarrow \\frac{F_0}{2}$**\n",
    "\n",
    "En discret pour prolonger la suite bornée (a, b, c), il y a deux choix à faire à chaque borne :\n",
    "* a est à l'instant $0$ et donc non répété : ..., b,a,b, ...\n",
    "* a est répété et donc à l'instant $\\frac{T_e}{2}$ : ..., b, a, a, b\n",
    "* d est non répété et donc à $(N-1).T_e$ de l'échantillon a : ..., c, d, c \n",
    "* d est répété et donc à l'instant $(N-1).T_e + \\frac{T_e}{2}$ de a : ..., c, d, d, c\n",
    "\n",
    "> en discret il y a 4 prolongements pairs donnant 4 **Direct Cosine Transform** DCT  \n",
    "> car il y a 4 prolongements pairs \"aux deux bornes\" conduisant à 4 espaces pairs périodiques,\n",
    "> et 4 prolongements pairs en \"a\" et impair en \"b\"\n",
    "\n",
    "\n",
    "Type de DCT | Prolongement (N=4)         | Décalage de \"a\" | parité en \"b\" | Période  \n",
    "-----|----------------------------|-----------------|---------------|--------\n",
    "Type-I | (a, b, c, b)       |         0       | pair          | 2.(N-1)   \n",
    "Type-II | (a, b, c, c, b, a) |        +1/2     | pair          | 2.N   \n",
    "\n",
    "Le type II est très utilisé pour compresser les signaux (mp3, avi, etc.) \n",
    "Donnons-en la base temporelle.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "366bb19a-d2f9-41d1-9cb7-0063cbcf1141",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "decal = 1/2;\n",
    "N = 3;  % 3 points\n",
    "j = (1:N) - decal;\n",
    "k = (0:8*N-1)' - decal;\n",
    "periode = 2*N;\n",
    "\n",
    "M_Bz_Bdct = peigne(k-j,periode) + ...\n",
    "            peigne(k-(periode-j),periode) ;\n",
    "\n",
    "\n",
    "for id_j=1:N\n",
    "    subplot(5,1,id_j); stem(k,M_Bz_Bdct(:,id_j),couleurs(id_j))\n",
    "end\n",
    "\n",
    "v_Bz = M_Bz_Bdct * v;\n",
    "subplot(5,1,[4 5]); \n",
    "stem(k,v_Bz,'k--');\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5876be88-fb3d-4db0-9c77-4000db9ad6b2",
   "metadata": {
    "tags": []
   },
   "source": [
    "Ce prolongement pair et périodique va permettre de décomposer le signal uniquement en cosisnus (a(n)) mais avec une résolution fréquentielle double, car la période est double."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17a29165-d304-4f53-a8f9-cd39e987acc6",
   "metadata": {
    "tags": []
   },
   "source": [
    "## Prolongement dans $\\mathbb{R}^{N}_{DCT-I}$ : suites paires périodiques (superpositions)\n",
    "\n",
    "Pour obtenir un décalage pair centré en $t=0$, il faut raccourcir la période à $2(N-1)$ et superposer les signaux :\n",
    "\n",
    ">  $(1,2,3)_W = \\vec{v} = 1.\\vec{w_0} + 2.\\vec{w_1} + 3.\\vec{w_3}  \\quad \\leftrightarrow \\quad  f: t\\mapsto 1.p[k] + 2.p[k-1] + 3.p[k-2]\\quad \\; | \\;\\forall k, f[k]=f[k+2(N-1)]$ et $f[k]=f[-k]$  \n",
    ">  avec la suite $(\\dots,\\, 3, 2,\\; 1, 2, 3, 2, \\; 1, 2, 3,\\,\\dots)$ par exemple   \n",
    "> **Pas de décalage temporel mais la période est $2.(N-1)$** car on superpose les extrémités\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fdddb1d8-4939-45c7-87f2-c2b69dc39d22",
   "metadata": {},
   "source": [
    "\n",
    "Dans les types de prolongement donnés\n",
    "\n",
    "\n",
    "Type de DCT | Prolongement (N=4)         | Décalage de \"a\" | parité en \"b\" | Période  \n",
    "-----|----------------------------|-----------------|---------------|--------\n",
    "Type-I | (a, b, c, b)       |         0       | pair          | 2.(N-1)   \n",
    "Type-II | (a, b, c, c, b, a) |        +1/2     | pair          | 2.N   \n",
    "\n",
    "Le type I nécessite de normer les vecteurs "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "7a959056-37c1-4e89-beed-386ecb9241e3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "decal = 0;\n",
    "N = 3;  % 3 points\n",
    "j = (0:(N-1)) - decal;\n",
    "k = (0:7*(N-1)-1)' - decal;\n",
    "periode = 2*(N-1);\n",
    "\n",
    "M_Bz_BdctI = peigne(k-j,periode) + ...\n",
    "            peigne(k-(periode-j),periode) ;\n",
    "\n",
    "\n",
    "for id_j=1:N\n",
    "    subplot(5,1,id_j); stem(k,M_Bz_BdctI(:,id_j),couleurs(id_j))\n",
    "end\n",
    "\n",
    "v_Bz = M_Bz_BdctI * v;\n",
    "subplot(5,1,[4 5]); \n",
    "stem(k,v_Bz,'k--');\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "319db787-d572-432f-9ffc-b7ea0d745d98",
   "metadata": {},
   "source": [
    "Ce prolongement de période \"trop courte\" recouvre la borne a et c :\n",
    "\n",
    "=  | 2c| b | 2a | b | 2c |  b | 2a | b\n",
    "---|---|---|---|---|---|---|---|---\n",
    "  +|---|---|---|---|---|---|---|---\n",
    "  +| c | b | a | 0 | 0 | 0 | 0 | 0 \n",
    "  +| 0 |0  | a | b | c | 0 | 0 | 0 \n",
    "  +| 0 |0  | 0 | 0 | c | b | a | 0 \n",
    "  +| 0 |0  | 0 | 0 | 0 | 0 | a | b \n",
    "  +|---|---|---|---|---|---|---|---\n",
    "\n",
    "Le \"a\" est doublé et le \"c\" aussi, car ainsi :\n",
    "- la base temporelle ..., (2,0,0,0), ... de a\n",
    "- la base temporelle ..., (0,0,2,0), ... de c\n",
    "- celles aux milieux ..., (0,1,0,1), ... de b\n",
    "ne sont pas de même norme !\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "9f6b092c-7440-44e6-9755-81b1c0a7414e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "normes =\n",
      "\n",
      "   2.0000   1.4142   2.0000\n",
      "\n",
      "correction =\n",
      "\n",
      "   0.50000   1.00000   0.50000\n",
      "\n",
      "normes =\n",
      "\n",
      "   1.0000   1.4142   1.0000\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "scalaires = scalp(M_Bz_BdctI,M_Bz_BdctI,2*(N-1));\n",
    "normes  = sqrt(sum(scalaires))\n",
    "correction = [0.5 1 0.5]\n",
    "% broadcast rangées\n",
    "M_Bz_BdctI = M_Bz_BdctI .* (correction);\n",
    "\n",
    "scalaires = scalp(M_Bz_BdctI,M_Bz_BdctI,2*(N-1));\n",
    "normes  = sqrt(sum(scalaires))\n",
    "\n",
    "for id_j=1:N\n",
    "    subplot(5,1,id_j); stem(k,M_Bz_BdctI(:,id_j),couleurs(id_j))\n",
    "end\n",
    "v_Bz = M_Bz_BdctI * v;\n",
    "subplot(5,1,[4 5]); \n",
    "stem(k,v_Bz,'k--');\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2178e32a-274a-4d89-99c3-042d48b920e0",
   "metadata": {},
   "source": [
    "Toujours pas de même norme !\n",
    "\n",
    "Dans le passage à une base fréquentielle il faudra traiter les points bornes à part ! \n",
    "\n",
    "Alors que si nous normons et acceptons de déformer un peu le signal, nous aurons une dct facile...\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "08dff993-13ff-4d4f-a4f3-ae92e1daaae3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "normes =\n",
      "\n",
      "   1.0000   1.4142   1.0000\n",
      "\n",
      "correction =\n",
      "\n",
      "   1.00000   0.70711   1.00000\n",
      "\n",
      "normes =\n",
      "\n",
      "   1.00000   1.00000   1.00000\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "scalaires = scalp(M_Bz_BdctI,M_Bz_BdctI,2*(N-1));\n",
    "normes  = sqrt(sum(scalaires))\n",
    "correction = 1./normes\n",
    "% broadcast rangées\n",
    "M_Bz_BdctI = M_Bz_BdctI .* (correction);\n",
    "\n",
    "scalaires = scalp(M_Bz_BdctI,M_Bz_BdctI,2*(N-1));\n",
    "normes  = sqrt(sum(scalaires))\n",
    "\n",
    "for id_j=1:N\n",
    "    subplot(5,1,id_j); stem(k,M_Bz_BdctI(:,id_j),couleurs(id_j))\n",
    "end\n",
    "v_Bz = M_Bz_BdctI * v;\n",
    "subplot(5,1,[4 5]); \n",
    "stem(k,v_Bz,'k--');\n"
   ]
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "31f2aee4e71d21fbe5cf8b01ff0e069b9275f58929596ceb00d14d90e3e16cd6"
  },
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "4.2.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}