RespostaImpulsoContinuo.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "88cd5f75-700a-46e4-b2b4-cbcd8401f3bd",
   "metadata": {},
   "source": [
    "# Resposta ao impulso de tempo contínuo\n",
    "\n",
    "Renato Naville Watanabe"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53bf38cf-ac2e-4ae8-9b3c-1be75aaa5044",
   "metadata": {},
   "source": [
    "## Preparação do ambiente"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "81a89813-934c-4297-8b9c-4ea708114b5d",
   "metadata": {},
   "outputs": [],
   "source": [
    "try:\n",
    "  from google.colab import output\n",
    "  output.enable_custom_widget_manager()\n",
    "  IN_COLAB = True\n",
    "except:\n",
    "  IN_COLAB = False\n",
    "\n",
    "if IN_COLAB: \n",
    "  !pip install ipympl\n",
    "  from IPython.display import clear_output \n",
    "  !sudo apt update\n",
    "  !sudo apt install libcairo2-dev ffmpeg \\\n",
    "       texlive texlive-latex-extra texlive-fonts-extra \\\n",
    "       texlive-latex-recommended texlive-science \\\n",
    "       tipa libpango1.0-dev\n",
    "  !pip install manim\n",
    "  !pip install IPython --upgrade\n",
    "  clear_output()\n",
    "  # Se estiver no Colab, reiniciar o ambiente após executar esta célula "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "cd9e5e23-ef5a-4e25-ac53-47dbbe9e5aa6",
   "metadata": {},
   "outputs": [
    {
     "ename": "ImportError",
     "evalue": "/home/rnwatanabe/miniconda3/lib/python3.10/site-packages/cairo/_cairo.cpython-310-x86_64-linux-gnu.so: undefined symbol: cairo_tee_surface_index",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mImportError\u001b[0m                               Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[3], line 2\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;21;01msympy\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m \u001b[38;5;21;01msym\u001b[39;00m\n\u001b[0;32m----> 2\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01mmanim\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m\n\u001b[1;32m      3\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01mmanim\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mutils\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mcolor\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Colors\n\u001b[1;32m      4\u001b[0m config\u001b[38;5;241m.\u001b[39mmedia_width \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m100\u001b[39m\u001b[38;5;124m%\u001b[39m\u001b[38;5;124m\"\u001b[39m\n",
      "File \u001b[0;32m~/miniconda3/lib/python3.10/site-packages/manim/__init__.py:26\u001b[0m\n\u001b[1;32m     24\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01manimation\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01manimation\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m\n\u001b[1;32m     25\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01manimation\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mchanging\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m\n\u001b[0;32m---> 26\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01manimation\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mcomposition\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m\n\u001b[1;32m     27\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01manimation\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mcreation\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m\n\u001b[1;32m     28\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01manimation\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mfading\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m\n",
      "File \u001b[0;32m~/miniconda3/lib/python3.10/site-packages/manim/animation/composition.py:16\u001b[0m\n\u001b[1;32m     14\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mconstants\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m RendererType\n\u001b[1;32m     15\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mmobject\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mmobject\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Group, Mobject\n\u001b[0;32m---> 16\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mscene\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mscene\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Scene\n\u001b[1;32m     17\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mutils\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01miterables\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m remove_list_redundancies\n\u001b[1;32m     18\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mutils\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mrate_functions\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m linear\n",
      "File \u001b[0;32m~/miniconda3/lib/python3.10/site-packages/manim/scene/scene.py:38\u001b[0m\n\u001b[1;32m     36\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m config, logger\n\u001b[1;32m     37\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01manimation\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01manimation\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Animation, Wait, prepare_animation\n\u001b[0;32m---> 38\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mcamera\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mcamera\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Camera\n\u001b[1;32m     39\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mconstants\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m\n\u001b[1;32m     40\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mgui\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mgui\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m configure_pygui\n",
      "File \u001b[0;32m~/miniconda3/lib/python3.10/site-packages/manim/camera/camera.py:16\u001b[0m\n\u001b[1;32m     13\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01mfunctools\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m reduce\n\u001b[1;32m     14\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01mtyping\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Any, Callable, Iterable\n\u001b[0;32m---> 16\u001b[0m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;21;01mcairo\u001b[39;00m\n\u001b[1;32m     17\u001b[0m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;21;01mnumpy\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m \u001b[38;5;21;01mnp\u001b[39;00m\n\u001b[1;32m     18\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01mPIL\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Image\n",
      "File \u001b[0;32m~/miniconda3/lib/python3.10/site-packages/cairo/__init__.py:1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01m_cairo\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;241m*\u001b[39m  \u001b[38;5;66;03m# noqa: F401,F403\u001b[39;00m\n\u001b[1;32m      4\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mget_include\u001b[39m():\n\u001b[1;32m      5\u001b[0m \u001b[38;5;250m    \u001b[39m\u001b[38;5;124;03m\"\"\"Returns a path to the directory containing the C header files\"\"\"\u001b[39;00m\n",
      "\u001b[0;31mImportError\u001b[0m: /home/rnwatanabe/miniconda3/lib/python3.10/site-packages/cairo/_cairo.cpython-310-x86_64-linux-gnu.so: undefined symbol: cairo_tee_surface_index"
     ]
    }
   ],
   "source": [
    "import sympy as sym\n",
    "from manim import *\n",
    "from manim.utils.color import Colors\n",
    "config.media_width = \"100%\"\n",
    "config.verbosity = \"WARNING\""
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fed8542a-c3c9-481e-97c3-3373962fbca8",
   "metadata": {},
   "source": [
    "## Definição\n",
    "\n",
    "A **resposta ao impulso** (às vezes também chamada de **resposta impulsiva**) de um sistema contínuo é o sinal de saída de um sistema quando um impulso $\\delta(t)$ é usado como sinal de entrada do sistema e inicialmente o sistema está em repouso (a saída $y(t)$ e todas as suas derivadas iguais a zero). A resposta ao impulso é normalmente representada pela letra $h(t)$.\n",
    "\n",
    "<img src=Imagens/diagimpulsocont.png width=\"500\">"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c10b8020-6b23-44ff-8550-24aadd44eba4",
   "metadata": {},
   "source": [
    "O sinal impulso é um sinal que ocorre em apenas um instante que tem amplitude infinita. É como se desse uma \"pancada\" no sistema ou o sistema fosse atingido por um raio. É algo muito forte que acontece por apenas um instante. Após o instante em que o impulso acontece ($t=0$) a entrada fica zero novamente.\n",
    "\n",
    "Mas nesse único instante, o sistema recebe uma quantidade de energia que muda toda a configuração do sistema, mudando assim os valores do sinal de saída e de suas derivadas. Desta forma é como se as condições iniciais do sistema mudassem devido ao impulso. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "072d34c2-f27e-426c-acb8-8d162de54733",
   "metadata": {},
   "source": [
    "Portanto, para encontrar a resposta ao impulso de um sistema contínuo, linear e invariante no tempo, com a seguinte forma geral de equação diferencial:\n",
    "\n",
    "$$\\frac{d^Ny}{dt^N} + a_{N-1}\\frac{d^{N-1}y}{dt^{N-1}}+...+a_0y(t) = b_N\\frac{d^Nx}{dt^N} +...+b_2\\frac{d^2x}{dt^2}+b_1\\frac{dx}{dt} + b_0x(t) $$\n",
    "\n",
    "basta encontrar a resposta de entrada nula do sistema considerando as novas condições iniciais devido ao impulso. Aqui esse sinal terá o símbolo $w(t)$ (é um sinal intermediário no cálculo da resposta ao impulso, sem nenhum significado físico). Essas condições iniciais são:\n",
    "\n",
    "$$\\frac{d^{N-1}w}{dt^{N-1}}(0) = 1, \\frac{d^{N-2}w}{dt^{N-2}}(0) = 0, ..., w(0) = 0$$\n",
    "\n",
    "Uma vez encontrada esse sinal $w(t)$, para encontrar a resposta ao impulso basta fazer:\n",
    "\n",
    "$$h(t)= \\left(b_N\\frac{d^Nw}{dt^N} +...+b_2\\frac{d^2w}{dt^2}+b_1\\frac{dw}{dt} + b_0w(t)\\right)u(t) $$\n",
    "\n",
    "\n",
    "Essa expressão de $h(t)$ é válida para $t>0$. Em $t=0$, se $b_N\\neq 0$ aparecerá um impulso com área $b_N$, somado a expresão anterior. \n",
    "\n",
    "$$h(t)= b_N\\delta(t) + \\left(b_N\\frac{d^Nw}{dt^N} +...+b_2\\frac{d^2w}{dt^2}+b_1\\frac{dw}{dt} + b_0w(t)\\right)u(t) $$\n",
    "\n",
    "\n",
    "A demonstração desse método está no vídeo abaixo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "id": "6146ba47-8f28-4a6d-a211-dbacfd05ed2e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "                                                                                                                       \r"
     ]
    },
    {
     "data": {
      "text/html": [
       "<video src=\"media/jupyter/ImpulseObtain@2023-04-28@13-04-03.mp4\" controls autoplay loop style=\"max-width: 100%;\"  >\n",
       "      Your browser does not support the <code>video</code> element.\n",
       "    </video>"
      ],
      "text/plain": [
       "<IPython.core.display.Video object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%manim -qm ImpulseObtain\n",
    "\n",
    "def step(x):\n",
    "    if x >=0: \n",
    "        return 1\n",
    "    else:\n",
    "        return 0   \n",
    "\n",
    "def zeros(x):\n",
    "    return 0   \n",
    "\n",
    "\n",
    "class ImpulseObtain(Scene):\n",
    "    def construct(self):\n",
    "        text1=MathTex(\n",
    "            \"\\\\frac{d^Ny}{dt^N} + a_{N-1}\\\\frac{d^{N-1}y}{dt^{N-1}}+a_{N-2}\\\\frac{d^{N-2}y}{dt^{N-2}}+...+a_0y(t)\", \n",
    "            \"=\", \n",
    "            \"b_M\\\\frac{d^Mx}{dt^M} +...+b_2\\\\frac{d^2x}{dt^2}+b_1\\\\frac{dx}{dt} + b_0x(t) \",\n",
    "        font_size=35)\n",
    "        self.play(Write(text1))\n",
    "        text2=MathTex(\n",
    "            \"\\\\frac{d^Ny}{dt^N} + a_{N-1}\\\\frac{d^{N-1}y}{dt^{N-1}}+a_{N-2}\\\\frac{d^{N-2}y}{dt^{N-2}}+...+a_0y(t)\", \n",
    "            \"=\", \n",
    "            \"w(t)\",        \n",
    "        font_size=35)\n",
    "        \n",
    "        text3=MathTex(\n",
    "            \"w(t)\", \n",
    "            \"=\", \n",
    "            \"b_M\\\\frac{d^Mx}{dt^M} +...+b_2\\\\frac{d^2x}{dt^2}+b_1\\\\frac{dx}{dt} + b_0x(t) \",        \n",
    "        font_size=35).move_to(DOWN)\n",
    "        text5=MathTex(\n",
    "            \"\\\\frac{d^Nw}{dt^N} + a_{N-1}\\\\frac{d^{N-1}w}{dt^{N-1}}+a_{N-2}\\\\frac{d^{N-2}w}{dt^{N-2}}+...+a_0w(t)\", \n",
    "            \"=\", \n",
    "            \"\\\\delta(t)\",        \n",
    "        font_size=35).move_to((0,3,0))\n",
    "        \n",
    "        text6=MathTex(\n",
    "            \"h(t)\", \n",
    "            \"=\", \n",
    "            \"b_M\\\\frac{d^Mw}{dt^M} +...+b_2\\\\frac{d^2w}{dt^2}+b_1\\\\frac{dw}{dt} + b_0w(t) \",        \n",
    "        font_size=35).move_to((0,2,0))\n",
    "        brace1 = Brace(text1[2])\n",
    "        brace1text = brace1.get_tex(\"w(t)\")\n",
    "        S1text = MathTex('S_1:  ').move_to((-5,0,0))\n",
    "        S2text = MathTex('S_2:  ').move_to((-5,-1,0))\n",
    "        # framebox2 = SurroundingRectangle(text[2], buff = .1)\n",
    "        # self.play(\n",
    "        #     Create(framebox1),\n",
    "        # )\n",
    "        self.wait()\n",
    "        self.add(brace1)\n",
    "        self.wait()\n",
    "        self.add(brace1text)\n",
    "        self.wait()\n",
    "        self.remove(brace1, brace1text)\n",
    "        self.play(\n",
    "            ReplacementTransform(text1,text2),\n",
    "            Write(text3)\n",
    "        )\n",
    "        self.add(S1text, S2text)\n",
    "        self.wait(2)\n",
    "        S1 = MathTex('S1').move_to((-5,0,0))\n",
    "        S2 = MathTex('S2').move_to((-5,-1,0))\n",
    "        self.play(S1.animate.move_to((3,-2,0)),\n",
    "                  S2.animate.move_to((-3,-2,0)),\n",
    "                  S1text.animate.move_to((-5,3,0)),\n",
    "                  S2text.animate.move_to((-5,2,0)), \n",
    "                  text2.animate.move_to((0,3,0)),\n",
    "                  text3.animate.move_to((0,2,0)))\n",
    "        rec1 = SurroundingRectangle(S1, buff=0.7)\n",
    "        rec2 = SurroundingRectangle(S2, buff=0.7)\n",
    "        line1 = Arrow([-2.05,-2,0], [2,-2,0], buff=0)\n",
    "        rec3 = SurroundingRectangle(Group(rec1, rec2, line1), buff=0.5)\n",
    "        line2 = Arrow([-6,-2,0], [-4,-2,0], buff=0)\n",
    "        line3 = Arrow([4,-2,0], [6,-2,0], buff=0)\n",
    "        x = MathTex('x(t)').move_to((-5,-1.5,0))\n",
    "        y = MathTex('y(t)').move_to((5,-1.5,0))\n",
    "        w = MathTex('w(t)').move_to((0,-1.5,0))\n",
    "        delta = MathTex('\\\\delta(t)').move_to((-5,-1.5,0))\n",
    "        h = MathTex('h(t)').move_to((5,-1.5,0))\n",
    "        self.add(rec1, rec2, line1, line2, line3, x, y, w, rec3)\n",
    "        self.wait(2)\n",
    "        self.play(ReplacementTransform(x, delta),\n",
    "                  ReplacementTransform(y, h))\n",
    "        self.wait(2)\n",
    "        text4 = Text('SLIT').move_to((0,0,0))\n",
    "        self.add(text4)\n",
    "        self.play(S1.animate.move_to((-3,-2,0)),\n",
    "                  S2.animate.move_to((3,-2,0)),\n",
    "                  ReplacementTransform(text2, text5),\n",
    "                  ReplacementTransform(text3, text6)\n",
    "                  )\n",
    "        self.wait(2)\n",
    "        sis = Group(line1, line2, line3, S1, S2, rec1, rec2, rec3, delta, h, w)\n",
    "        S2group = Group(S2text, text6)\n",
    "        \n",
    "        self.remove(text4)\n",
    "        self.play(sis.animate.scale(0.5), S2group.animate.scale(0.7))\n",
    "        self.play(sis.animate.move_to((-4,-3,0)), \n",
    "                  S2group.animate.move_to((2.5, -3,0)))\n",
    "        \n",
    "        \n",
    "        ax = Axes(x_range=[-1, 4], y_range=[-1, 5], tips=False).scale(0.2).move_to((-6, 1.5,0))\n",
    "        line4 = Arrow([-6.7,1.1,0], [-6.7,2,0], buff=0, color=BLUE)\n",
    "        \n",
    "        ax2 = Axes(x_range=[-1, 4], y_range=[-1, 5], tips=False).scale(0.2).move_to((-3, 1.5,0))\n",
    "        graph2 = ax2.plot(lambda x: step(x), color=BLUE, x_range=[-1, 0.5, 0.01],\n",
    "                        use_smoothing=False)\n",
    "        \n",
    "        ax3 = Axes(x_range=[-1, 4], y_range=[-1, 5], tips=False).scale(0.2).move_to((0, 1.5,0))\n",
    "        graph3 = ax3.plot(lambda x: zeros(x), color=BLUE, x_range=[-1, 0.5, 0.01],\n",
    "                            use_smoothing=False)\n",
    "        \n",
    "        ax4 = Axes(x_range=[-1, 4], y_range=[-1, 5], tips=False).scale(0.2).move_to((2.7, 1.5,0))\n",
    "        graph4 = ax4.plot(lambda x: zeros(x), color=BLUE, x_range=[-1, 0.5, 0.01],\n",
    "                        use_smoothing=False)\n",
    "        \n",
    "        ax5 = Axes(x_range=[-1, 4], y_range=[-1, 5], tips=False).scale(0.2).move_to((6, 1.5,0))\n",
    "        line5 = Arrow([5.3,1.1,0], [5.3,2,0], buff=0, color=BLUE)\n",
    "        \n",
    "        text6 = MathTex('+').move_to((-4.8,1.5,0))\n",
    "        text7 = MathTex('+').move_to((-1.8,1.5,0))\n",
    "        text8 = MathTex('...').move_to((1.2,1.5,0))\n",
    "        text9 = MathTex('=').move_to((4.2,1.5,0))\n",
    "        self.add(ax, line4, ax2, graph2, ax3, graph3, \n",
    "                 ax4, graph4, ax5, line5, \n",
    "                 text6, text7, text8, text9)\n",
    "        self.wait(2)\n",
    "        text10 = MathTex('\\\\frac{d^Nw}{dt^N}(0) = \\\\delta(0)').next_to(ax, DOWN).scale(0.7)\n",
    "        text11 = MathTex('\\\\frac{d^{N-1}w}{dt^{N-1}}(0) = 1').next_to(ax2, DOWN).scale(0.7)\n",
    "        text12 = MathTex('\\\\frac{d^{N-2}w}{dt^{N-2}}(0) = 0').next_to(ax3, DOWN).scale(0.7)\n",
    "        text13 = MathTex('w(0) = 0').next_to(ax4, DOWN).scale(0.7)\n",
    "        self.add(text10, text11, text12, text13)   \n",
    "        winit = Group(text10, text11, text12, text13)\n",
    "        self.wait(5)\n",
    "        self.remove(ax, line4, ax2, graph2, ax3, graph3, \n",
    "                    ax4, graph4, ax5, line5, text6, \n",
    "                    text7, text8, text9)\n",
    "        text13 =  MathTex('\\\\frac{d^Nw}{dt^N}(0) = \\\\delta(0), ', '\\\\frac{d^{N-1}w}{dt^{N-1}}(0) = 1, ',\n",
    "                          '\\\\frac{d^{N-2}w}{dt^{N-2}}(0) = 0, ','...', 'w(0) = 0').scale(0.6).next_to(text5, DOWN)   \n",
    "        self.play(ReplacementTransform(winit, text13))\n",
    "        self.wait(3)\n",
    "        text14 = MathTex('w(t) = y_{zi}(t), \\\\text{ com } ', '\\\\frac{d^{N-1}w}{dt^{N-1}}(0) = 1, ',\n",
    "                          '\\\\frac{d^{N-2}w}{dt^{N-2}}(0) = 0, ','...', 'w(0) = 0').scale(0.6).next_to(text5, DOWN)\n",
    "        self.play(ReplacementTransform(text13, text14))\n",
    "        self.wait(2)\n",
    "        S1group = Group(S1text, text5)\n",
    "        self.play(S2group.animate.scale(1/0.7), S1group.animate.scale(0.7))\n",
    "        self.play(S2group.animate.move_to((-2, 3,0)),\n",
    "                  S1group.animate.move_to((2.5, -3, 0)))\n",
    "        \n",
    "        self.wait(2)\n",
    "        text15=MathTex(\n",
    "            \"h(t)\", \n",
    "            \"=\", \n",
    "            \"b_M\\\\frac{d^Mw}{dt^M} +...+b_2\\\\frac{d^2w}{dt^2}+b_1\\\\frac{dw}{dt} + b_0w(t) \",        \n",
    "        font_size=35).move_to((-1.5,3,0))\n",
    "        self.remove(text14)\n",
    "        self.play(ReplacementTransform(S2group, text15))\n",
    "        self.wait(5)\n",
    "        \n",
    "        \n",
    "        \n",
    "        \n",
    "        "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d300155d-6aee-4577-961e-a2d3748a1132",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "### Exemplo 1:\n",
    "\n",
    "Vamos encontrar a resposta ao impulso para o sistema com a equação diferencial abaixo:\n",
    "\n",
    "$\\frac{dy}{dt} - 0.5y(t) = x(t)$\n",
    "\n",
    "A raiz do polinômio característico da equação diferencial é $\\lambda_1=0.5$. Então $w(t)$ é:\n",
    "\n",
    "$w(t)=A_1e^{\\lambda t}$\n",
    "\n",
    "Considerando a condição inicial $w(0) = 1$.\n",
    "\n",
    "$w(t) = e^{0.5 t}$\n",
    "\n",
    "Portanto, a resposta ao impulso desse sistema é:\n",
    "\n",
    "$h(t) = e^{0.5 t}u(t)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "593a3f19-5b52-4234-9904-92fc6902fcf8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fce3e0479a0>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sym.symbols('t')\n",
    "sym.plot(sym.exp(0.5*t), (t,0, 6))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69e4f6cd-5600-4fb5-ab12-09381bc9258f",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6fc0e626-b29f-450f-a90b-dcd7b8194147",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "### Exemplo 2\n",
    "\n",
    "Vamos agora obter uma expressão da resposta ao impulso da seguinte equação diferencial:\n",
    "\n",
    "$\\frac{d^2y}{dt^2} + 3\\frac{dy}{dt} - 4y(t) = 3x(t) + \\frac{dx}{dt}$\n",
    "\n",
    "As raízes do polinômio característico da equação são:\n",
    "\n",
    "$\\lambda_1 = \\frac{-3 + \\sqrt{9+16}}{2} = 1$ e $\\lambda_2 = \\frac{-3 - \\sqrt{9+16}}{2} = -4$\n",
    "\n",
    "O sinal $w(t)$ é:\n",
    "\n",
    "$w(t) = A_1e^{t} + A_2e^{-4t}$, com $w(0) = 0$ e $\\frac{dw}{dt}(0) = 1$.\n",
    "\n",
    "Então:\n",
    "\n",
    "$w(t) = 0.2e^t-0.2e^{-4t}$\n",
    "\n",
    "Usando os coeficientes que multiplicam $x(t)$ e suas derivadas, temos:\n",
    "\n",
    "$$h(t) = (3w(t) + \\frac{dw}{dt})u(t) = (0.6e^t-0.6e^{-4t} + 0.2e^t +0.8e^{-4t})u(t) = (0.8e^t+0.2e^{-4t})u(t)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "c1728608-d9c0-4799-85a7-617561868358",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fce3dfd3280>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sym.symbols('t')\n",
    "sym.plot(0.8*sym.exp(t) + 0.2*sym.exp(-4*t), (t,0, 6))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2b9efbd6-fbf5-421d-adb1-f68dd5be2da4",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3d6cb78-8bb3-4f77-b8a1-009a95e0df64",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "### Exemplo 3\n",
    "\n",
    "Vamos encontrar a resposta ao impulso de um sistema com a seguinte equação diferencial.\n",
    "\n",
    "$$\\frac{d^2y}{dt^2} = -4\\frac{dy}{dt} - 4y(t) + 3x(t)$$\n",
    "\n",
    "O polinômio característico é:\n",
    "\n",
    "$\\lambda^2 + 4\\lambda + 4 = 0$\n",
    "\n",
    "As raízes da equação são iguais e iguais a -2. Então, o sinal $w(t)$ é:\n",
    "\n",
    "$w(t) = c_1e^{-2t} + c_2te^{-2t}$, com $w(0) = 0$ e $\\frac{dw}{dt}(0) = 1$.\n",
    "\n",
    "Então, \n",
    "\n",
    "$$w(t) = te^{-2t}$$\n",
    "\n",
    "Portanto , a resposta ao impulso é:\n",
    "\n",
    "$$h(t) = 3te^{-2t}u(t)$$\n",
    "\n",
    "O gráfico da resposta ao impulso está abaixo:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "24c2b850-562a-4fd7-9772-8c4f877bc2d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fce3e0beb90>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sym.symbols('t')\n",
    "sym.plot(3*sym.exp(-2*t)*t, (t,0, 6))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2edaf618-d95d-4058-988c-499f7436b374",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a87563c-d35f-400e-8641-9e2fff2bb8cc",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "### Exemplo 4\n",
    "\n",
    "Considere um sistema com a seguinte equação diferencial:\n",
    "\n",
    "$$\\frac{d^2y}{dt^2} = -2\\frac{dy}{dt} - 2y(t) + x(t) + 2\\frac{dx}{dt}$$\n",
    "\n",
    "Para encontrar a resposta ao impulso desse sistema, calculamos as raízes do seu polinômio característico.\n",
    "\n",
    "$\\lambda_1 = \\frac{-2 + \\sqrt{4-8}}{2} = -1+j$ e $\\lambda_2 = \\frac{-2 - \\sqrt{4-8}}{2} = -1-j$\n",
    "\n",
    "Como as raízes são complexas, o sinal $w(t)$ terá a forma:\n",
    "\n",
    "$w(t) = Ae^{-t}\\cos(t+\\theta)$, com $w(0) = 0$ e $\\frac{dw}{dt}(0) = 1$.\n",
    "\n",
    "Usando os valores iniciais, temos:\n",
    "\n",
    "$w(t) = e^{-t}\\cos\\left(t-\\frac{\\pi}{2}\\right)$\n",
    "\n",
    "Portanto, a resposta ao impulso é:\n",
    "\n",
    "$$h(t) = w(t) + 2\\frac{dw}{dt} = e^{-t}\\cos\\left(t-\\frac{\\pi}{2}\\right) - 2e^{-t}\\cos\\left(t-\\frac{\\pi}{2}\\right) - 2e^{-t}\\mathrm{sen}\\left(t-\\frac{\\pi}{2}\\right) = - e^{-t}\\cos\\left(t-\\frac{\\pi}{2}\\right) - 2e^{-t}\\mathrm{sen}\\left(t-\\frac{\\pi}{2}\\right)$$\n",
    "\n",
    "O gráfico está mostrado abaixo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8b4cb030-4c51-43e1-88cb-fdba2f4c7762",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = sym.symbols('t')\n",
    "sym.plot(-sym.exp(-t)*(sym.cos(t-sym.pi/2)+2*sym.sin(t-sym.pi/2)), (t,0, 6));\n",
    "        "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64120b6e-005d-4256-903a-b4178af3c7ad",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc8b5ebb-81f3-4d11-94ab-cb3a19704c54",
   "metadata": {},
   "source": [
    "## Exercícios\n",
    "\n",
    "**1)** Encontre a resposta ao impulso dos sistemas descritos pelas seguintes equações diferenciais e condições iniciais.\n",
    "\n",
    "a) $\\frac{d^2y}{dt^2} = -4\\frac{dy}{dt}-6y(t) + \\frac{dx}{dt} + x(t)$, com $y(0) = 2, \\frac{dy}{dt}(0) = 1$\n",
    "\n",
    "b) $\\frac{d^2y}{dt^2} = -4\\frac{dy}{dt}-4y(t)  + x(t)$, com $y(0) = 3, \\frac{dy}{dt}(0) = -4$\n",
    "\n",
    "c) $\\frac{d^2y}{dt^2} = -\\frac{dy}{dt} + \\frac{dx}{dt} + x(t)$, com $y(0) = 1, \\frac{dy}{dt}(0) = 1$\n",
    "\n",
    "d) $\\frac{d^2y}{dt^2} = -4y(t) + x(t)$, com $y(0) = 0, \\frac{dy}{dt}(0) = 6$\n",
    "\n",
    "e) $\\frac{d^2y}{dt^2} = -4\\frac{dy}{dt}-13y(t) + 4\\frac{dx}{dt} + 2x(t)$, com $y(0) = 1, \\frac{dy}{dt}(0) = 9.9$\n",
    "\n",
    "f) $\\frac{d^3y}{dt^3} = -1\\frac{d2y}{dt^2} + 2\\frac{d^2x}{dt^2} + 3x(t)$, com $y(0) = 2, \\frac{dy}{dt}(0) = 1, \\frac{d2y}{dt^2}(0) = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50b02979",
   "metadata": {},
   "source": [
    "**2)** Considere o circuito elétrico da figura abaixo:\n",
    "\n",
    "<img src=https://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/RC_Series_Filter_%28with_V%26I_Labels%29.svg/1920px-RC_Series_Filter_%28with_V%26I_Labels%29.svg.png width=\"400\">\n",
    "\n",
    "Considere que o sinal de entrada do sistema é a tensão do gerador $x(t) = V_{in}$ e o sinal de saída é a tensão no resistor $y(t)=V_R$. Pede-se:\n",
    "\n",
    "a) A equação diferencial que descreve o comportamento desse sistema (lembre-se que $I=C\\frac{dV_c}{dt}$, $I = \\frac{V_r}{R}$ e $V_{in} = V_c + V_r$).\n",
    "\n",
    "b) A resposta ao impulso desse sistema."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6df607c3-ed44-4957-9c5f-02f23b4819cc",
   "metadata": {},
   "source": [
    "**3)** Quais destas respostas ao impulso são de sistemas causais. Todos os sistemas são lineares e invariantes no tempo:\n",
    "\n",
    "a) $h_1(t) = e^{-2|t|}$\n",
    "\n",
    "b) $h_2(t) = (1-|t|)[u(t+2)-u(t-2)]$\n",
    "\n",
    "c) $h_3(t) = e^{3t}u(-t)$\n",
    "\n",
    "d) $h_4(t) = e^{3t}u(t)$\n",
    "\n",
    "e) $h_5(t) = \\cos(3t)u(t)$\n",
    "\n",
    "f) $h_6(t) = \\frac{t}{t+1}u(t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c690a46b-60ca-40fd-85cd-0dfab838ef85",
   "metadata": {},
   "source": [
    "**4)** Encontre a resposta ao impulso dos sistemas descritos pelas seguintes equações diferenciais:\n",
    "\n",
    "a) $\\frac{d^4 y}{dt^4} = -y(t) + x(t)$, com  com $y(0)=1$, $\\frac{dy}{dt}(0) = 1 $, $\\frac{d^2y}{dt^2}(0) = 0$, $\\frac{d^3y}{dt^3}(0) = -1 $\n",
    "\n",
    "b) $\\frac{d^3 y}{dt^3} = -y(t) + x(t) + \\frac{dx}{dt}$, com  com $y(0)=1$, $\\frac{dy}{dt}(0) = 1 $, $\\frac{d^2y}{dt^2}(0) = 0$\n",
    "\n",
    "c) $\\frac{d^4 y}{dt^4} = -\\frac{1}{16}y(t) + x(t)$, com  com $y(0)=1$, $\\frac{dy}{dt}(0) = 1 $, $\\frac{d^2y}{dt^2}(0) = 0$, $\\frac{d^3y}{dt^3}(0) = -1 $\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85098a5a-e4e7-4170-8a11-c8fcd98fe9c4",
   "metadata": {},
   "source": [
    "## Referências\n",
    "\n",
    "- Lathi, BP, Sinais e Sistemas Lineares, 2007"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "abbf5709-d251-48c1-89bd-4170a087324b",
   "metadata": {},
   "source": [
    "## Solução dos exercícios"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4137e0f7-0a97-4e02-923c-e6d5abaf7c7b",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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