{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "5ba69335-23fd-4749-83a5-b076f9348ca4",
   "metadata": {},
   "source": [
    "Ejercicio 14 - Serie de Fourier - Item a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "4f3816aa-6648-44e4-b6c5-ec02e71b3f41",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<StemContainer object of 3 artists>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# desarrollo la seria de fourier de x[n] con periodo N = 12\n",
    "k = np.array(range(12))\n",
    "ak = (1/12) * (1 + 2 * np.cos((np.pi/6) * k))\n",
    "plt.stem(k, ak)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "7a5d2afb-f08a-4846-9db0-a5ece36a6f96",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<StemContainer object of 3 artists>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# corroboramos que la serie de fourier esta bien calculada\n",
    "# para eso vamos a definir una funcion que nos reconstruya la señal a partir de su serie de fourier\n",
    "\n",
    "def reconstruir_senial(n, a) :\n",
    "    x = np.zeros_like(n).astype(complex)\n",
    "    N = len(a)\n",
    "    for k in range(N) :\n",
    "        x += a[k] * np.exp(1j * ((2*np.pi)/N) * k * n)\n",
    "    return np.real(x)\n",
    "\n",
    "# obtengo la señal x[n]\n",
    "\n",
    "n = np.array(range(-20,20))\n",
    "x = reconstruir_senial(n, ak)\n",
    "plt.stem(n,x)  \n",
    "        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "a696e914-260e-465d-b2e6-b49383da66ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<StemContainer object of 3 artists>"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# desarrollo la seria de fourier de y[n-3] con periodo N = 6 (version centrada en el origen)\n",
    "\n",
    "k = np.array(range(6))\n",
    "bk = (1/6) * (1 + 2 * np.cos((np.pi/3) * k))\n",
    "plt.stem(k, bk)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "25f1e9f7-cf3e-4ffc-bd00-00b1a0a2d612",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<StemContainer object of 3 artists>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# para calcular la serie de y[n] tengo que desplazar 3 a derecha, lo que implica multiplicar a los coeficientes\n",
    "# por exp(-j * ((2*pi))/6 * k * 3)\n",
    "\n",
    "ck = bk * np.exp(-1j * ((2*np.pi)/6) * k * 3)\n",
    "plt.stem(k, ck)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "d39a7a39-60b0-4c3a-8d0b-e35e03003fcf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<StemContainer object of 3 artists>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# obtengo la señal y[n] a partir de su serie de fourier\n",
    "\n",
    "y = reconstruir_senial(n, ck)\n",
    "plt.stem(n,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "5fb358bb-edbc-46df-97cb-b22e94311682",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<StemContainer object of 3 artists>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# me interesa tener un desarrollo de y[n] con periodo N = 12 de modo de poder comparar coeficientes en la entrada vs coeficientes\n",
    "# en la salida\n",
    "# para pasar de un desarrollo en N = 6 a un desarrollo en N = 12 aparecen ceros en la serie de fourier\n",
    "\n",
    "k = np.array(range(12))\n",
    "dk = np.zeros_like(k).astype(complex)\n",
    "for i in range(6) :\n",
    "    dk[2*i] = ck[i]\n",
    "\n",
    "plt.stem(k, dk)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "fe7faa43-e743-403a-b616-3c767ce9e82c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<StemContainer object of 3 artists>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# reconstruimos la señal corroborar que venimos bien\n",
    "\n",
    "y = reconstruir_senial(n, dk)\n",
    "plt.stem(n, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "4a061283-3ed8-40e6-9f5e-3e71fc74cd10",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# muy bien!, ahora si podemos hacer una comparacion entre los coeficientes de x[n] e y[n]\n",
    "\n",
    "plt.stem(k, ak, linefmt='b-', label='serie de x[n]', markerfmt='bo', basefmt='gray')\n",
    "plt.stem(k, dk, linefmt='r-', label='serie de y[n]', markerfmt='ro', basefmt='gray')\n",
    "plt.legend() \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1332d3c7-341e-4d7e-97fd-a5074be92e5b",
   "metadata": {},
   "outputs": [],
   "source": [
    "# notar que siempre que tenemos un armonico en la salida (rojo) hay un armonico asociado en la entrada, lo que quiere decir\n",
    "# que el sistema no produjo nuevos componentes\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc551c4d-57e9-485c-90bc-ea5a98162dc0",
   "metadata": {},
   "source": [
    "Para el item b) hay que hacer lo mismo. Los desarrollos deben hacerse en N = 36 para permitir la comparacion de armonicos de entrada y salida"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.11.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}