{ "cells": [ { "cell_type": "markdown", "id": "688473bf-df40-4e0d-9123-c72ece5dfb26", "metadata": {}, "source": [ "# Analyse spectrale numérique" ] }, { "cell_type": "code", "execution_count": 1, "id": "b9bb4e1e-56e7-4f3f-a057-bcf0af47844f", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from numpy.fft import fft\n", "\n", "plt.rcParams['figure.figsize'] = [10, 6] # taille de sortie du graphique (modifiable)\n", "plt.rcParams['axes.titlesize'] = 12 # taille caractères titre du graphique (modifiable)" ] }, { "cell_type": "markdown", "id": "8d52fec9-d877-441e-a212-6ccf0394d610", "metadata": {}, "source": [ "La fonction **spectre** prend comme arguments :\n", "+ la fonction **signal** définissant le signal ;\n", "+ la fréquence d'échantillonnage **F_ech** ;\n", "+ la durée totale **duree** de l'échantillon ;\n", "+ un dernier argument optionnel qui indique la fréquence maximale jusqu'à laquelle est tracé le spectre.\n", "\n", "Elle retourne le spectre du signal." ] }, { "cell_type": "code", "execution_count": 5, "id": "1cdf9e51-07f9-4756-8a5d-a0273d094801", "metadata": {}, "outputs": [], "source": [ "def spectre(signal,F_ech,duree,*args):\n", " t = np.arange(0,duree,1/F_ech) # on construit le tableau des instants d'échantillonnage\n", " ech = signal(t) # on construit l'échantillon\n", " tfd = fft(ech) # on calcule le spectre par FFT\n", " N = len(tfd) # on construit le tableau des fréquences \n", " freq = np.zeros(N)\n", " for k in range(N):\n", " freq[k] = 1./duree*k\n", " spectre = np.absolute(tfd)*2./N # norme des composantes du spectre, en normalisant\n", " spectre[0] = .5*spectre[0] # normalisation de la composante continue\n", " if len(args)>0: # argument optionnel : on trace le spectre\n", " f_max = args[0] # jusqu'à la fréquence f_max\n", " else:\n", " f_max = F_ech\n", " N_max = int(N*f_max/F_ech)\n", " plt.stem(freq[:N_max],spectre[:N_max],markerfmt=\" \",basefmt=\" \") # tracé du spectre\n", " plt.xlabel('fréquence')\n", " plt.ylabel('amplitude')\n", " plt.title('spectre calculé par FFT')\n", " plt.show()\n", " return" ] }, { "cell_type": "markdown", "id": "b66de54d-3340-4946-a993-6b7e339efd25", "metadata": {}, "source": [ "Exemple : $s(t)=0,5+\\cos(2\\pi t)+0,6\\cos(3\\pi 2t)$" ] }, { "cell_type": "code", "execution_count": 3, "id": "16e8c95a-99cb-407c-b45d-d673f9aabf95", "metadata": {}, "outputs": [], "source": [ "def s(t):\n", " return .5 + np.cos(2*np.pi*t) + .6*np.cos(2*np.pi*3*t)" ] }, { "cell_type": "markdown", "id": "4011a8a9-2d02-4593-b48f-9dd811105bc0", "metadata": {}, "source": [ "Le spectre doit être constitué de:\n", "+ une composante continue (fréquence nulle) d'amplitude 0,5 ;\n", "+ une composante de fréquence $f_1=1$ d'amplitude 1 ;\n", "+ une composante de fréquence $f_2=3$ d'amplitude 0,6." ] }, { "cell_type": "code", "execution_count": 9, "id": "f0720bb1-49a0-4498-8e55-cfb13a110ace", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "spectre(s,20,10) # la fréquence d'échantillonnage est 20" ] }, { "cell_type": "markdown", "id": "44a1ee9c-f698-4490-9939-a2f688d0217e", "metadata": {}, "source": [ "On retourve bien les trois composantes du spectre, avec leurs amplitudes respectives. \n", "On observe la duplication du spectre, symmétrique par rapport à $f_\\mathrm{e}/2=10$.\n", "Le critère de Shannon est bien respecté.\n", "\n", "On peut ne représenter que la partie utile du spectre, jusqu'à $f_\\mathrm{e}/2$ :" ] }, { "cell_type": "code", "execution_count": 10, "id": "110cf936-acc7-4990-a20a-88c06d4f95b6", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "spectre(s,20,10,10)" ] }, { "cell_type": "markdown", "id": "51f16740-b6c6-405f-b536-71b6cb3f3111", "metadata": {}, "source": [ "Prenons comme fréquence d'échantillonnage $F_\\mathrm{ech}=5$.\n", "La condition de Shannon n'est pas respectée : il faut $F_\\mathrm{ech}>2f_\\mathrm{max}=2\\times{}3=6$.\n", "On représente la partie \"utile\" du spectre, sur l'intervalle $[0,F_\\mathrm{ech}/2]$." ] }, { "cell_type": "code", "execution_count": 11, "id": "79ed1bd4-4a4a-43ca-81a1-c47fcd6af240", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "spectre(s,5,10,2.5)" ] }, { "cell_type": "markdown", "id": "4f52bc81-2420-4b62-8166-df53ab619aaa", "metadata": {}, "source": [ "Il apparaît dans le spectre la fréquence $f=2$, qui est absente du signal étudié. C'est le phénomène de **repliement de spectre** : on retrouve le symétrique de chaque fréquence du signal par rapport à $F_\\mathrm{ech}/2=2,5$. La fréquence $f_2=3$ donne donc une raie à la fréquence $2$, symétrique par rapport à $2,5$. \n", "\n", "On peut mieux le comprendre en visualisant la totalité du spectre :" ] }, { "cell_type": "code", "execution_count": 12, "id": "3cd4c161-7bb7-4040-811b-1fee70c195ff", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "spectre(s,5,10)" ] }, { "cell_type": "markdown", "id": "8bab1ea3-baf4-4a4a-8ac6-e7b48bfac3b0", "metadata": {}, "source": [ "Le signal a pour période $T=1$. Nous avons précédemment calculé le spectre pour une durée totale d'échantillon $T_\\mathrm{tot}=10$, multiple de la période $T$ du signal.\n", "\n", "Que se passe-t-il si la durée de l'échantillon n'est plus un multiple de la période du signal ?" ] }, { "cell_type": "code", "execution_count": 13, "id": "2805d902-6201-44d1-b2a4-c6a154f20dc8", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "spectre(s,10,10.5,5) # F_ech = 10, tracé sur [0,F_ech/2]" ] }, { "cell_type": "markdown", "id": "e60599e5-e4a3-41e5-9444-dd10cc569daf", "metadata": {}, "source": [ "Le spectre n'est pas celui attendu.!\n", "\n", "Le calcul de la FFT se base sur la construction d'un signal périodique, à partir de l'échantillon de longueur finie (donc non périodique), en concaténant des copies de l'échantillon. On obtient alors un signal de période maximale $T_\\mathrm{ech}$.\n", "\n", "Si la durée de l'échantillon est un multiple de celle du signal initial, le signal construit s'identifie au signal initial. Si ce n'est pas le cas, on a construit un signal différent, dont le spectre est logiquement différent de celui attenndu.\n", "\n", "On peut \"limiter les dégats\" en multipliant le signal par une fenêtre : on montre que les choses s'améliorent si le signal périodique construit est continu aux points de raccordement. On multiplie donc l'échantillon par une fonction qui tend vers 0 aux extrémités (la fenêtre), assurant la continuité du raccordement.\n", "\n", "Il existe plusieurs types de fenêtres, accessibles avec numpy. On les sélectionne ici selon :\n", "\n", "1 : fenêtre de Hamming,\n", "2 : fenêtre de Hanning,\n", "3 : fenêtre de Blakcman,\n", "4 : fenêtre de Bartlett,\n", "5 : fenêtre de Kaiser,\n", "autre : sans fenêtre" ] }, { "cell_type": "code", "execution_count": 15, "id": "954a8175-7ca1-4e8d-95f6-87ee1e3fbc06", "metadata": {}, "outputs": [], "source": [ "def spectre_fenetre(signal,F_ech,duree,fenetre,*args):\n", " t = np.arange(0,duree,1/F_ech) # on construit le tableau des instants d'échantillonnage\n", " ech = signal(t) # on construit l'échantillon\n", " M = len(ech) # nombre de points de l'échantillon\n", " if fenetre == 1:\n", " window = np.hamming(M) # fenêtre de Hamming\n", " nom = \"avec fenêtre de Hamming\"\n", " elif fenetre == 2:\n", " window = np.hanning(M) # fenêtre de Hanning \n", " nom = \"avec fenêtre de Hanning\"\n", " elif fenetre == 3:\n", " window = np.blackman(M) # fenêtre de Blackman \n", " nom = \"avec fenêtre de Blackman\"\n", " elif fenetre == 4:\n", " window = np.bartlett(M) # fenêtre de Bartlett\n", " nom = \"avec fenêtre de Bartlett\"\n", " elif fenetre == 5:\n", " window = np.kaiser(M,14) # fenêtre de Kaiser (coeff. modifiable)\n", " nom = \"avec fenêtre de Kaiser\"\n", " else:\n", " window = 1.\n", " nom = \"sans fenêtre\"\n", " ech_fenetre = window*ech\n", " tfd = fft(ech_fenetre) # on calcule le spectre par FFT\n", " N = len(tfd) # on construit le tableau des fréquences \n", " freq = np.zeros(N)\n", " for k in range(N):\n", " freq[k] = 1./duree*k\n", " spectre = np.absolute(tfd)*2./N # norme des composantes du spectre, en normalisant\n", " spectre[0] = .5*spectre[0] # normalisation de la composante continue\n", " if len(args)>0: # argument optionnel : on trace le spectre\n", " f_max = args[0] # jusqu'à la fréquence f_max\n", " else:\n", " f_max = F_ech\n", " N_max = int(N*f_max/F_ech)\n", " plt.stem(freq[:N_max],spectre[:N_max],markerfmt=\" \",basefmt=\" \") # tracé du spectre\n", " plt.xlabel(\"fréquence\")\n", " plt.ylabel(\"amplitude\")\n", " plt.title(\"spectre calculé par FFT \"+str(nom))\n", " plt.show\n", " return" ] }, { "cell_type": "code", "execution_count": 16, "id": "ab0690a3-7b56-41ed-b86a-dd0b38df7039", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "spectre_fenetre(s,10,10.5,1,5)" ] }, { "cell_type": "markdown", "id": "385c25de-b0c9-4bef-99c2-35f8c8a227d6", "metadata": {}, "source": [ "On retrouve un spectre plus proche du spectre attendu, avec trois pics, certes un peu élargis." ] }, { "cell_type": "code", "execution_count": null, "id": "ad93820e-8174-44d2-9d43-ec0936f0d970", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.8" } }, "nbformat": 4, "nbformat_minor": 5 }