conv_eq_recursiva.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](imgs/enunciado3.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](imgs/enunciado_3.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Base teórica\n",
    "Para fazer transformada Z e conversão de eq. recursiva é fundamental usar a propriedade de time-shifting da transformada Z."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](imgs/tz_time_shift.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para encontrar as transformadas envolvidas foi utilizado [essa tabela](https://lpsa.swarthmore.edu/LaplaceZTable/LaplaceZFuncTable.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vamos de resolução"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](imgs/transf_z_3_2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**OBS**: Importante notar que a aplicação da propriedade de time-shift foi para equivaler os termos aos da tabela, por conta disso que a transformada de y(k+1) foi representada como Z*Y(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Piecewise((25*UnitImpulse(n) + 75/4 - 175*5**(-n)/4, n >= 0))\n"
     ]
    }
   ],
   "source": [
    "from lcapy.discretetime import z\n",
    "Yz=(10*z+5)/((z-1)*(z-0.2))\n",
    "xk=Yz.IZT()\n",
    "print(xk) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "from matplotlib.pyplot import plot\n",
    "from numpy import arange, ones, zeros\n",
    "\n",
    "k = arange(kmax+1);\n",
    "y_ex = 75/4 - (175/4)*(1/5**(k));\n",
    "y_ex[0] += 25;\n",
    "\n",
    "kmax = 20;\n",
    "y_rc = zeros(kmax+1);\n",
    "e = 15*ones(kmax+1);\n",
    "e[0] = 10;\n",
    "\n",
    "for i in range(1, kmax+1):\n",
    "    y_rc[i] = 0.2*y_rc[i-1] + e[i-1];\n",
    "    \n",
    "plot(k, y_rc, \"o\");\n",
    "plot(k, y_ex, \"+\");"
   ]
  }
 ],
 "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}