clases/clase_7/700_sistemas.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sistemas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Índice "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Objetivos del notebook "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Clasificación de sistemas\n",
    "\n",
    "En este contexto, un sistema puede considerarse como un proceso en el cual las señales de entrada son transformadas por el sistema o provocan que éste responda de alguna forma, lo que da como resultado otras señales de salida. Por ejemplo, un sistema de alta fidelidad toma una señal de audio grabada y genera una reproducción de dicha señal. Si el sistema de alta fidelidad tiene controles de tono, podemos cambiar la calidad en el matiz de la señal reproducida.\n",
    "Un sistema continuo es aquel en el cual las señales continuas de entrada son transformadas en señales continuas de salida:\n",
    "\n",
    "$$x(t) \\rightarrow y(t)$$\n",
    "\n",
    "De la misma manera, un sistema discreto, es uno que transforma señales de entrada discretas en salidas de tiempo discreto:\n",
    "\n",
    "$$x[n] \\rightarrow y[n]$$\n",
    "\n",
    "¿Algún sistema que combine ambos dominios?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Propiedades basicas de un sistema\n",
    "\n",
    "Vamos a repasar las 4 propiedades basicas de un sistema continuo, las mismas se aplican a sistemas discretos [Kamen, Edward W., 2008]:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Sistemas sin y con memoria\n",
    "\n",
    "Un sistema es **sin memoria**, si su salida para cada valor de la variable independiente en un tiempo dado depende solamente de la entrada en ese instante de tiempo.\n",
    "\n",
    "$$S:y[n]=(2x[x]-x^2[n])^2$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "x = np.arange(0,101)\n",
    "y = (2*x-x**2)**2\n",
    "\n",
    "plt.plot(y, '.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Un sistema **con memoria** posee algun mecanismo en el sistema que almacena información sobre los valores de entrada en instantes anteriores (y/o futuro) al tiempo actual.\n",
    "\n",
    "$$S:y[n]=y[n-1]+x[n]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.ones(100);  \n",
    "\n",
    "def S(x):\n",
    "    y[0]=0+x[0]\n",
    "    for n in range(1,len(x)):\n",
    "        y[n] = y[n-1]+x[n]\n",
    "    return y\n",
    "\n",
    "S(x)\n",
    "plt.subplot(2,1,1)\n",
    "plt.plot(x, '.r')\n",
    "plt.subplot(2,1,2)\n",
    "plt.plot(y, '.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Invertibilidad y sistemas inversos\n",
    "\n",
    "Si un sistema es **inversible** entonces existe el sistema inverso tal que cuando esta en cascada con el sistema original produce una salida w que es igual a la entrada. Un ejemplo de sistema inversible y su inversa es:\n",
    "\n",
    "$$S_1:y[n]=\\sum_{k=0}^{n}x[k]$$\n",
    "\n",
    "$$S_2 = S_1^{-1}$$\n",
    "\n",
    "$$S_2:w[n]=y[n]-y[n-1]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.ones(100)\n",
    "\n",
    "def S1(x):\n",
    "    for n in range(len(x)):\n",
    "        y[n] = np.sum(x[0:n+1])\n",
    "    return y\n",
    "\n",
    "def S2(y):\n",
    "    w = np.zeros(len(y))\n",
    "    w[0] = y[0]-0 # indice problematico y[-1]=0\n",
    "    for n in range(1,len(y)):\n",
    "        w[n] = y[n]-y[n-1]\n",
    "    return w\n",
    "\n",
    "plt.subplot(3,1,1)\n",
    "plt.plot(x, '.r')\n",
    "plt.subplot(3,1,2)\n",
    "plt.plot(S1(x), '.')\n",
    "plt.subplot(3,1,3)\n",
    "plt.plot(S2(S1(x)), '.g')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Un ejemplo de sistema no inversible es:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fdec5ab15d0>]"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.signal as signal\n",
    "\n",
    "T = 1000;\n",
    "Fs = 10000;\n",
    "t = np.linspace(0, T, Fs)\n",
    "x1 = np.sin(2*np.pi*50*t);\n",
    "x2 = signal.sawtooth(2*np.pi*50*t+np.pi);  # diente de sierra\n",
    "y1 = np.sign(x1);  # funcion signo\n",
    "y2 = np.sign(x2);\n",
    "\n",
    "plt.subplot(4,1,1)\n",
    "plt.plot(x1,'r')\n",
    "plt.subplot(4,1,2)\n",
    "plt.plot(x2,'r')\n",
    "plt.subplot(4,1,3)\n",
    "plt.plot(y1,'r')\n",
    "plt.subplot(4,1,4)\n",
    "plt.plot(y2,'r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Causalidad\n",
    "\n",
    "Un sistema es **causal** si su salida en cualquier instante de tiempo depende sólo de los valores de entrada en el momento presente y en el pasado. También se los llama no anticipativo. Todos los sistemas sin memoria son causales ya que la salida responde sólo a valores presentes de la entrada y/o salida. Un ejemplo de sistema causal.\n",
    "\n",
    "$$S_1:y[n]=\\sum_{k=0}^{n}x[k]$$\n",
    "\n",
    "$$S_2:y[n]=x[n]+x[n-2]$$\n",
    "\n",
    "$$S_3:y[n]=7x[n-5]$$\n",
    "\n",
    "Dado que el sistema causal no incluye muestras de entrada futuras; dicho sistema es prácticamente realizable. Eso significa que dicho sistema puede implementarse prácticamente. En general, todos los sistemas de tiempo real son sistemas causales; porque en aplicaciones en tiempo real solo están presentes muestras presentes y pasadas. Dado que las muestras futuras no están presentes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fdec61a2810>]"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ACenidXfxmqGb193Fa4YRXnfr59wlSWcroblLks5guEtSgVod7hFxXUT8ICIeioj9kx7POETEZRHx1Yh4ICK+FxE31ccvjoivRMQP6+1Fkx7rqEXEBRHxrYj4Yr1/eUQcru/35+pfTFeUiNgcEXdFxPcj4sGI2NmRe/3O+s/3/RHx2Yh4Zmn3OyI+EREnI+L+vmMD721UPlZf+3ci4qr1vl9rw73v4/xeAVwJvCYirpzsqMbiFPCuzLwSuBp4S32d+4GlzLwCWKr3S3MT8GDf/oeAj2bmHwE/A/ZOZFTjdSvw5cz8Y+BFVNdf9L2OiG3A24G5zHwh1S8cfDXl3e9PAdedcWy1e/sK4Ir6ax9w23rfrLXhTt/H+WXmCtD7OL+iZOaJzPxm/f0vqf5j30Z1rQfq0w4AfzGZEY5HRGwHXgncXu8HcC1wV31Kidf8POBlwB0AmbmSmU9R+L2ubQKeFRGbgAuBExR2vzPza8B/n3F4tXu7G/h0Vr4ObI6Iret5vzaH+6CP89s2obFsiIjYAbwYOAxsycwT9UOPA1smNKxxWQDeDfxfvX8J8FRmnqr3S7zflwNPAp+sp6Nuj4hnU/i9zszjwIeBR6lC/efAUcq/37D6vW2cb20O906JiOcAXwDekZm/6H8sq/WsxaxpjYhXAScz8+ikx7LBNgFXAbdl5ouB/+GMKZjS7jVAPc+8m+p/br8PPJuzpy+KN+p72+Zw78zH+UXE06mC/TOZeXd9+IneX9Pq7clJjW8MrgH+PCJ+QjXddi3VXPTm+q/tUOb9PgYcy8zD9f5dVGFf8r0GeDnw48x8MjN/DdxN9Weg9PsNq9/bxvnW5nDvxMf51XPNdwAPZuZH+h66F9hTf78HuGejxzYumfmezNyemTuo7uu/Z+brgK8CN9SnFXXNAJn5OPBYRLygPrQLeICC73XtUeDqiLiw/vPeu+6i73dttXt7L/CGetXM1cDP+6Zv1iYzW/sFXA/8F/Aj4G8nPZ4xXeOfUv1V7TvAt+uv66nmoJeAHwL/Blw86bGO6frngS/W3/8h8J/AQ8DngWdMenxjuN4/AY7U9/ufgIu6cK+BvwO+D9wP/CPwjNLuN/BZqn9T+DXV39L2rnZvgaBaDfgj4LtUK4nW9X7++gFJKlCbp2UkSasw3CWpQIa7JBXIcJekAhnuklQgw12SCmS4S1KB/h+hRKVhjNb3FwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.ones(100)\n",
    "\n",
    "def S1(x):\n",
    "    for n in range(len(x)):\n",
    "        y[n] = np.sum(x[0:n+1])\n",
    "    return y\n",
    "\n",
    "plt.plot(S1(x),'.r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Un sistema cuya respuesta presente depende de los valores futuros de las entradas se llama como un sistema **no causal**.\n",
    "\n",
    "![](../img/no_causales.jpg)\n",
    "\n",
    "$$S_1:y[n]=x[n+1]$$\n",
    "\n",
    "$$S_2:y[n]=x[n]+x[n-2]$$\n",
    "\n",
    "$$S_3:y[n]=7x[n+5]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fab75eec290>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAD4CAYAAADiry33AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+j8jraAAAT50lEQVR4nO3df6xcZ33n8fenThyWHyIJvsuythOnu1ZLCs2PjgJXoHJpSurQbkxFpXXKdtMKZKlKttCtdhW2UrMbpG27XQFLmxIs8Aaq4rRNoHURELKGu6GtQz2m2ZAfhJhAG1vp+haHwBaEa+e7f5zjauLcmzu25/raj98vaTRznuecme+ZY3/uuc+cuU+qCklSu75vuQuQJC0tg16SGmfQS1LjDHpJapxBL0mNO2u5C5jPqlWrat26dctdhiSdNnbv3v13VTU1X98pGfTr1q1jOBwudxmSdNpI8tcL9Tl0I0mNM+glqXEGvSQ1zqCXpMYZ9JLUuEWDPsnaJJ9L8lCSB5O8fZ51kuR9SfYkuT/J5SN91yV5tL9dN+kdkCQ9t3EurzwE/EpVfTHJi4DdSe6uqodG1rkaWN/fXgW8H3hVkvOBm4ABUP2226vqyYnuhSSdjnbuhNlZmJmB6ekle5lFg76qngCe6B9/O8nDwGpgNOg3Ah+p7m8e35vk3CQvA2aAu6vqAECSu4ENwLaJ7oUknW527oQrr4SDB2HlStixY8nC/pjG6JOsAy4DvnBU12rg8ZHlvX3bQu3zPffmJMMkw7m5uWMpS5JOP7OzXcgfPtzdz84u2UuNHfRJXgjcCbyjqr416UKqaktVDapqMDU177d4JakdMzPdmfyKFd39zMySvdRYfwIhydl0If/7VfWxeVbZB6wdWV7Tt+2jG74ZbZ89nkIlqSnT091wzakwRp8kwIeAh6vq3Qusth24IcntdB/GPlVVTyS5C/ivSc7r17sKeOcE6pak09/09JIG/BHjnNG/Bvg54EtJ7uvb/hNwAUBV3Qp8EngjsAf4DvALfd+BJO8CdvXb3Xzkg1lJ0skxzlU3fwZkkXUKuH6Bvq3A1uOqTpJ0wvxmrCQ1zqCXpMYZ9JLUOINekhpn0EtS4wx6SWqcQS9JjTPoJalxBr0kNc6gl6TGGfSS1DiDXpIaZ9BLUuMMeklqnEEvSY0z6CWpceNMJbgV+Clgf1W9Yp7+/wC8ZeT5Xg5M9bNLfR34NnAYOFRVg0kVLkkazzhn9LcBGxbqrKrfqqpLq+pSuvlg//dR0wW+vu835CVpGSwa9FV1DzDuPK/XAttOqCJJ0kRNbIw+yfPpzvzvHGku4DNJdifZvMj2m5MMkwzn5uYmVZYknfEm+WHsvwL+/Khhm9dW1eXA1cD1SX50oY2raktVDapqMDU1NcGyJOnMNsmg38RRwzZVta+/3w98HLhigq8nSRrDRII+yYuB1wF/MtL2giQvOvIYuAp4YBKvJ0ka3ziXV24DZoBVSfYCNwFnA1TVrf1qPw18pqr+fmTTlwIfT3LkdT5aVZ+eXOmSpHEsGvRVde0Y69xGdxnmaNtjwCXHW5gkaTL8ZqwkNc6gl6TGGfSS1DiDXpIaZ9BLUuMMeklqnEEvSY0z6CWpcQa9JDXOoJekxhn0ktQ4g16SGmfQS1LjDHpJapxBL0mNM+glqXGLBn2SrUn2J5l3GsAkM0meSnJff/u1kb4NSR5JsifJjZMsXJI0nnHO6G8DNiyyzuer6tL+djNAkhXALcDVwMXAtUkuPpFiJUnHbtGgr6p7gAPH8dxXAHuq6rGqOgjcDmw8jueRJJ2ASY3RTyf5P0k+leSH+rbVwOMj6+zt2+aVZHOSYZLh3NzchMqSJE0i6L8IXFhVlwC/Dfzx8TxJVW2pqkFVDaampiZQliQJJhD0VfWtqvp//eNPAmcnWQXsA9aOrLqmb5MknUQnHPRJ/lmS9I+v6J/zG8AuYH2Si5KsBDYB20/09SRJx+asxVZIsg2YAVYl2QvcBJwNUFW3Aj8D/GKSQ8B3gU1VVcChJDcAdwErgK1V9eCS7IUkaUHpMvnUMhgMajgcLncZknTaSLK7qgbz9fnNWElqnEEvSY0z6CWpcQa9JDXOoJekxhn0ktQ4g16SGmfQS1LjDHpJapxBL0mNM+glqXEGvSQ1zqCXpMYZ9JLUOINekhq3aNAn2Zpkf5IHFuh/S5L7k3wpyV8kuWSk7+t9+31J/APzkrQMxjmjvw3Y8Bz9XwNeV1WvBN4FbDmq//VVdelCfxBfkrS0Fp1KsKruSbLuOfr/YmTxXrpJwCVJp4hJj9G/FfjUyHIBn0myO8nm59owyeYkwyTDubm5CZclSWeuRc/ox5Xk9XRB/9qR5tdW1b4k/xS4O8mXq+qe+bavqi30wz6DweDUm8hWkk5TEzmjT/LDwAeBjVX1jSPtVbWvv98PfBy4YhKvJ0ka3wkHfZILgI8BP1dVXxlpf0GSFx15DFwFzHvljiRp6Sw6dJNkGzADrEqyF7gJOBugqm4Ffg14CfC7SQAO9VfYvBT4eN92FvDRqvr0EuyDJOk5jHPVzbWL9L8NeNs87Y8Blzx7C0nSyeQ3YyWpcQa9JDXOoJekxhn0ktQ4g16SGmfQS1LjDHpJapxBL0mNM+glqXEGvSQ1zqCXpMYZ9JLUOINekhpn0EtS4wx6SWqcQS9JjRsr6JNsTbI/ybxTAabzviR7ktyf5PKRvuuSPNrfrptU4ZKk8Sw6w1TvNuB3gI8s0H81sL6/vQp4P/CqJOfTTT04AArYnWR7VT15IkUvaOdOmJ2FmZlu+XR+PD3d1v5M6rHvi+9LS+/F9DQnw1hBX1X3JFn3HKtsBD5SVQXcm+TcJC+jm2v27qo6AJDkbmADsO1Eip7Xzp1w5ZVw8CCsWAEJHDp0ej5euRLe+154xzva2B/fF98X34v5a92x46SE/bhn9ItZDTw+sry3b1uo/VmSbAY2A1xwwQXHXsHsbHdgDx+Gp5/u2qpOz8cHD8Kdd7azP74vvi++F/PXOjt7WgX9CauqLcAWgMFgUMf8BDMz3U/IU/2n+Lg/6d/8Zvj859vYH98X3xffi/lrPTKUs8QmFfT7gLUjy2v6tn10wzej7bMTes1nmp7ufg1a7jG3ST2enoZXvnL56zjVHvu++L609F6cpDH6dMPqY6yYrAM+UVWvmKfvJ4EbgDfSfRj7vqq6ov8wdjdw5CqcLwI/cmTMfiGDwaCGw+G4+yBJZ7wku6tqMF/fWGf0SbbRnZmvSrKX7kqaswGq6lbgk3Qhvwf4DvALfd+BJO8CdvVPdfNiIS9Jmqxxr7q5dpH+Aq5foG8rsPXYS5MkTcL3LXcBkqSlZdBLUuMMeklqnEEvSY0z6CWpcQa9JDXOoJekxhn0ktQ4g16SGmfQS1LjDHpJapxBL0mNM+glqXEGvSQ1zqCXpMYZ9JLUuLGCPsmGJI8k2ZPkxnn635Pkvv72lSTfHOk7PNK3fZLFS5IWt+gMU0lWALcAbwD2AruSbK+qh46sU1W/PLL+vwMuG3mK71bVpZMrWZJ0LMY5o78C2FNVj1XVQeB2YONzrH8tsG0SxUmSTtw4Qb8aeHxkeW/f9ixJLgQuAj470vy8JMMk9yZ500IvkmRzv95wbm5ujLIkSeOY9Iexm4A7qurwSNuFVTUAfhZ4b5J/Md+GVbWlqgZVNZiamppwWZJ05hon6PcBa0eW1/Rt89nEUcM2VbWvv38MmOWZ4/eSpCU2TtDvAtYnuSjJSrowf9bVM0l+EDgP2DnSdl6Sc/rHq4DXAA8dva0kaeksetVNVR1KcgNwF7AC2FpVDya5GRhW1ZHQ3wTcXlU1svnLgQ8keZruh8pvjF6tI0laenlmLp8aBoNBDYfD5S5Dkk4bSXb3n4c+i9+MlaTGGfSS1DiDXpIaZ9BLUuMMeklqnEEvSY0z6CWpcQa9JDXOoJekxhn0ktQ4g16SGmfQS1LjDHpJapxBL0mNM+glqXFjBX2SDUkeSbInyY3z9P98krkk9/W3t430XZfk0f523SSLlyQtbtEZppKsAG4B3gDsBXYl2T7PTFF/UFU3HLXt+cBNwAAoYHe/7ZMTqV6StKhxzuivAPZU1WNVdRC4Hdg45vP/BHB3VR3ow/1uYMPxlSpJOh7jBP1q4PGR5b1929HenOT+JHckWXuM25Jkc5JhkuHc3NwYZUmSxjGpD2P/FFhXVT9Md9b+4WN9gqraUlWDqhpMTU1NqCxJ0jhBvw9YO7K8pm/7R1X1jar6Xr/4QeBHxt1WkrS0xgn6XcD6JBclWQlsAraPrpDkZSOL1wAP94/vAq5Kcl6S84Cr+jZJ0kmy6FU3VXUoyQ10Ab0C2FpVDya5GRhW1Xbgl5JcAxwCDgA/3297IMm76H5YANxcVQeWYD8kSQtIVS13Dc8yGAxqOBwudxmSdNpIsruqBvP1+c1YSWqcQS9JjTPoJalxBr0kNc6gl6TGGfSS1DiDXpIaZ9BLUuMMeklqnEEvSY0z6CWpcQa9JDXOoJekxhn0ktQ4g16SGmfQS1Ljxgr6JBuSPJJkT5Ib5+n/90keSnJ/kh1JLhzpO5zkvv62/ehtJUlLa9GpBJOsAG4B3gDsBXYl2V5VD42s9lfAoKq+k+QXgf8G/Ou+77tVdemE65YkjWmcM/orgD1V9VhVHQRuBzaOrlBVn6uq7/SL9wJrJlumJOl4jRP0q4HHR5b39m0LeSvwqZHl5yUZJrk3yZsW2ijJ5n694dzc3BhlSZLGsejQzbFI8m+AAfC6keYLq2pfku8HPpvkS1X11aO3raotwBboJgefZF2SdCYb54x+H7B2ZHlN3/YMSX4c+FXgmqr63pH2qtrX3z8GzAKXnUC9kqRjNE7Q7wLWJ7koyUpgE/CMq2eSXAZ8gC7k94+0n5fknP7xKuA1wOiHuJKkJbbo0E1VHUpyA3AXsALYWlUPJrkZGFbVduC3gBcCf5QE4G+q6hrg5cAHkjxN90PlN466WkeStMRSdeoNhw8GgxoOh8tdhiSdNpLsrqrBfH1+M1aSGmfQS1LjDHpJapxBL0mNM+glqXEGvSQ1zqCXpMYZ9JLUOINekhpn0EtS4wx6SWqcQS9JjTPoJalxBr0kNc6gl6TGGfSS1Lixgj7JhiSPJNmT5MZ5+s9J8gd9/xeSrBvpe2ff/kiSn5hc6ZKkcSwa9ElWALcAVwMXA9cmufio1d4KPFlV/xJ4D/Cb/bYX080x+0PABuB3++eTJI3auRN+/de7+wlbdM5Y4ApgT1U9BpDkdmAjz5zkeyPwn/vHdwC/k27y2I3A7VX1PeBrSfb0zzf5PZGk09XOnXDllXDwIKxcCTt2wPT0xJ5+nKGb1cDjI8t7+7Z516mqQ8BTwEvG3BaAJJuTDJMM5+bmxqteklowO9uF/OHD3f3s7ESf/pT5MLaqtlTVoKoGU1NTy12OJJ08MzPdmfyKFd39zMxEn36coZt9wNqR5TV923zr7E1yFvBi4BtjbitJZ7bp6W64Zna2C/kJDtvAeEG/C1if5CK6kN4E/OxR62wHrqMbe/8Z4LNVVUm2Ax9N8m7gnwPrgb+cVPGS1Izp6YkH/BGLBn1VHUpyA3AXsALYWlUPJrkZGFbVduBDwO/1H7YeoPthQL/eH9J9cHsIuL6qDi/JnkiS5pWqWu4anmUwGNRwOFzuMiTptJFkd1UN5us7ZT6MlSQtDYNekhpn0EtS4wx6SWrcKflhbJI54K+Pc/NVwN9NsJzTgfvcvjNtf8F9PlYXVtW83zY9JYP+RCQZLvTJc6vc5/adafsL7vMkOXQjSY0z6CWpcS0G/ZblLmAZuM/tO9P2F9zniWlujF6S9EwtntFLkkYY9JLUuGaCfrEJzFuQZG2SzyV5KMmDSd7et5+f5O4kj/b35y13rZOWZEWSv0ryiX75on4i+j39xPQrl7vGSUpybpI7knw5ycNJpls/zkl+uf93/UCSbUme19pxTrI1yf4kD4y0zXtc03lfv+/3J7n8eF+3iaAfcwLzFhwCfqWqLgZeDVzf7+eNwI6qWg/s6Jdb83bg4ZHl3wTe009I/yTdBPUt+R/Ap6vqB4FL6Pa92eOcZDXwS8Cgql5B9yfRN9Hecb4N2HBU20LH9Wq6OTzWA5uB9x/vizYR9IxMYF5VB4EjE5g3paqeqKov9o+/TfeffzXdvn64X+3DwJuWp8KlkWQN8JPAB/vlAD9GNxE9NLbPSV4M/CjdPA9U1cGq+iaNH2e6+TH+ST9L3fOBJ2jsOFfVPXRzdoxa6LhuBD5SnXuBc5O87Hhet5WgH3sS8lYkWQdcBnwBeGlVPdF3/S3w0mUqa6m8F/iPwNP98kuAb/YT0UN7x/siYA74n/1w1QeTvICGj3NV7QP+O/A3dAH/FLCbto/zEQsd14nlWitBf0ZJ8kLgTuAdVfWt0b7qrpdt5prZJD8F7K+q3ctdy0l0FnA58P6qugz4e44apmnwOJ9HdwZ7Ed20oy/g2UMczVuq49pK0J8xk5AnOZsu5H+/qj7WN//fI7/S9ff7l6u+JfAa4JokX6cbkvsxuvHrc/tf8aG9470X2FtVX+iX76AL/paP848DX6uquar6B+BjdMe+5eN8xELHdWK51krQ/+ME5v2n8pvoJixvSj82/SHg4ap690jXkcnZ6e//5GTXtlSq6p1Vtaaq1tEd189W1VuAz9FNRA/t7fPfAo8n+YG+6Uq6eZebPc50QzavTvL8/t/5kX1u9jiPWOi4bgf+bX/1zauBp0aGeI5NVTVxA94IfAX4KvCry13PEu3ja+l+rbsfuK+/vZFuzHoH8Cjwv4Dzl7vWJdr/GeAT/ePvB/4S2AP8EXDOctc34X29FBj2x/qPgfNaP87AfwG+DDwA/B5wTmvHGdhG9xnEP9D95vbWhY4rELqrCb8KfInuiqTjel3/BIIkNa6VoRtJ0gIMeklqnEEvSY0z6CWpcQa9JDXOoJekxhn0ktS4/w8vKXWhHonK0AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.ones(100)\n",
    "x = np.append(x,2)\n",
    "\n",
    "def S1(x):\n",
    "    for n in range(len(x)-1):\n",
    "        y[n] = x[n+1]\n",
    "    return y\n",
    "\n",
    "plt.plot(S1(x),'.r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dado que el sistema no causal contiene muestras futuras; un sistema no causal prácticamente no es realizable. Eso significa que en casos prácticos no es posible implementar un sistema no causal.\n",
    "Pero si las señales se almacenan en la memoria y posteriormente las utiliza un sistema, dichas señales se tratan como señales avanzadas o futuras. Debido a que tales señales ya están presentes, antes de que el sistema haya comenzado su operación. En tales casos, es posible implementar un sistema no causal.\n",
    "Algunos ejemplos prácticos de sistemas no causales son los siguientes:\n",
    "\n",
    "* Crecimiento de la población.\n",
    "\n",
    "* Pronóstico del tiempo.\n",
    "\n",
    "* Valores de acciones."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Estabilidad\n",
    "\n",
    "Sabemos que un procedimiento para determinar si **un sistema es estable**, si este es LTI. es comprobar que la respuesta impulsiva tiene suma acotada, sin embargo para sistemas no LTI debemos aplicar la definición general (cualquier entrada acotada produce una salida acotada) [Olivas, Emilio Soria, 2003]. En definitiva, un sistema es estable cuando perturbaciones pequeñas de la entrada conducen a respuestas que no divergen. Ejemplo de un **sistema inestable**:\n",
    "\n",
    "$$S_1:y[n]=nx[n]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fab756a5590>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.ones(100)\n",
    "\n",
    "def S1(x):\n",
    "    y = np.zeros(len(x))\n",
    "    for n in range(len(x)):\n",
    "        y[n] = n*x[n]\n",
    "    return y\n",
    "\n",
    "plt.plot(S1(x),'.r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Un sistema estable:\n",
    "\n",
    "$$S_2:y[n]=e^{x[n]}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fab7573a610>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.ones(100)\n",
    "\n",
    "def S1(x):\n",
    "    y = np.zeros(len(x))\n",
    "    for n in range(len(x)):\n",
    "        y[n] = np.exp(x[n])\n",
    "    return y\n",
    "\n",
    "plt.plot(S1(x),'.r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Invariancia en el tiempo\n",
    "\n",
    "Un sistema es **invariante en el tiempo** si el comportamiento y características del mismo están fijos en el tiempo. Un sistema es invariante con el tiempo si un desplazamiento temporal en la entrada ocasiona un despazamiento temporal en la salida."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$S_1:y[n]=sen[x[n]] \\qquad \\text{Verificar que es invariante en el tiempo.}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$S_2:y[n]=nx[n] \\qquad \\text{Verificar que es variante en el tiempo.}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Linealidad\n",
    "\n",
    "Un **sistema lineal**, tanto en tiempo continuo como discreto posee la importante propiedad de superposición es decir si una entrada es la suma ponderada de varias señales la salida es la superposición de las respuestas del sistema a cada una de estas señales.\n",
    "\n",
    "$$ ax_1[n] \\rightarrow ay_1[n] \\\\ bx_2[n] \\rightarrow by_2[n] \\\\ \\text{Entonces: } x_3=ax_1[n]+bx_2[n]\\rightarrow y_3=ay_1[n]+by_2[n] \\\\ $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fab73fa05d0>]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 100\n",
    "alfa = 2 \n",
    "beta = 0.5\n",
    "    \n",
    "x1 = np.arange(0,N)  # secuencia arbitraria n°1, para el estudio de la LINEALIDAD.\n",
    "x2 = np.sin(2*np.pi*0.3*np.arange(0,N))  # secuencia arbitraria n°2, para el estudio de la LINEALIDAD.\n",
    "x3 = alfa*x1 + beta*x2  # combinacion lineas de las secuencias n°1 y n°2, para el estudio de la LINEALIDAD.\n",
    "\n",
    "y1,y2,y3 = np.zeros(len(x)),np.zeros(len(x)),np.zeros(len(x))\n",
    "\n",
    "for n in range(len(x)):\n",
    "    y1[n] = n*x1[n] # n+x[n]\n",
    "    y2[n] = n*x2[n]\n",
    "    y3[n] = n*x3[n]\n",
    "\n",
    "plt.plot(y3,'ro')  \n",
    "plt.plot(alfa*y1+beta*y2,'g*')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Veamos un ejemplo para verificar la **linealidad**, la **estabilidad** e **invarianza temporal** de un sistema discreto: \n",
    "\n",
    "$$y[n] = x[n]-2x[n-1]-y[n-2]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Generamos 5 secuencia para estudiar el sistema\n",
    "N = 100;\n",
    "alfa = 2; beta = 0.5;\n",
    "    \n",
    "x1 = np.arange(0,N)  # secuencia arbitraria n°1, para el estudio de la LINEALIDAD.\n",
    "x2 = np.sin(2*np.pi*0.3*np.arange(0,N))  # secuencia arbitraria n°2, para el estudio de la LINEALIDAD.\n",
    "x3 = alfa*x1 + beta*x2  # combinacion lineas de las secuencias n°1 y n°2, para el estudio de la LINEALIDAD.\n",
    "\n",
    "x4 = np.concatenate((np.ones(1), np.zeros(N-1))) # secuencia para estudiar la ESTABILIDAD.\n",
    "x5 = np.concatenate((np.zeros(5), x1[1:N-4]))  # secuencia para estudiar la INVARIANZA TEMPORAL, retardando 5 muestras.\n",
    "\n",
    "# Consideramos las siguientes condiciones iniciales n = 0 --> x [n-1]=x[-1]= 0 ; y[n-2]=y[-2]= 0.5 // n = 1 --> y[n-2]=y[-1] = 2\n",
    "y_1 = 0; y_2 = 0; x_1 = 0;\n",
    "\n",
    "# Ahora, debemos calcular las salidas para las entradas anteriores y1[n],y2[n], y3[n], y4[n], por ultimo y5[n]   \n",
    "\n",
    "# Primero, determinamos fuera del bucle la salida para índices problemáticos n = 0 y n = 1\n",
    "\n",
    "y1,y2,y3,y4,y5 = np.zeros(N),np.zeros(N),np.zeros(N),np.zeros(N),np.zeros(N)\n",
    "\n",
    "# Para n = 0\n",
    "n = 0\n",
    "y1[n] = x1[0]-2*x_1-y_2\n",
    "y2[n] = x2[0]-2*x_1-y_2\n",
    "y3[n] = x3[0]-2*x_1-y_2\n",
    "y4[n] = x4[0]-2*x_1-y_2\n",
    "y5[n] = x5[0]-2*x_1-y_2\n",
    "\n",
    "# Para n = 1\n",
    "n = 1\n",
    "y1[n] = x1[1]-2*x1[0]-y_1\n",
    "y2[n] = x2[1]-2*x2[0]-y_1\n",
    "y3[n] = x3[1]-2*x3[0]-y_1\n",
    "y4[n] = x4[1]-2*x4[0]-y_1\n",
    "y5[n] = x5[1]-2*x5[0]-y_1\n",
    "\n",
    "# Realizamos los cálculos para n > 1\n",
    "for n in range(2,N):\n",
    "    y1[n] = x1[n]-2*x1[n-1]-y1[n-2]\n",
    "    y2[n] = x2[n]-2*x2[n-1]-y2[n-2]\n",
    "    y3[n] = x3[n]-2*x3[n-1]-y3[n-2]\n",
    "    y4[n] = x4[n]-2*x4[n-1]-y4[n-2]\n",
    "    y5[n] = x5[n]-2*x5[n-1]-y5[n-2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dibujamos las salidas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(*args, **kw)>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Linealidad\n",
    "plt.plot(y3,'ro')  \n",
    "plt.plot(alfa*y1+beta*y2,'g*')\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(*args, **kw)>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Estabilidad\n",
    "plt.plot(y4,'r')\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb46e5503d0>]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Invarianza temporal\n",
    "plt.plot(y5,'ro')\n",
    "plt.plot(np.concatenate((np.zeros(5), y1[1:N-4])),'g*')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://realpython.com/python-matplotlib-guide/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Referencias\n",
    "\n",
    " * Kamen, Edward W., and Bonnie S. Heck. Fundamentos de señales y sistemas usando la Web y Matlab. Pearson Prentice-Hall, 2008\n",
    " * Olivas, Emilio Soria. Tratamiento digital de señales: problemas y ejercicios resueltos. Pearson Educación, 2003"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Licencia\n",
    "\n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-sa/4.0/\"><img alt=\"Licencia de Creative Commons\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-sa/4.0/88x31.png\" /></a><br />Este documento se destribuye con una <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-sa/4.0/\">licencia Atribución CompartirIgual 4.0 Internacional de Creative Commons</a>.\n",
    "\n",
    "Material adaptado del curso [Introducción a Python](https://github.com/infiniemlabs-acustica/python_introduccion)\n",
    "\n",
    "© 2020. Infiniem Labs Acústica. infiniemlab.dsp@gmail.com (CC BY-SA 4.0))"
   ]
  }
 ],
 "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": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}