IntegralConvolução.ipynb

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   "source": [
    "# Integral de Convolução\n",
    "\n",
    "Renato Naville Watanabe"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b336592-ea2a-4860-b0e0-278e2482bdda",
   "metadata": {},
   "source": [
    "## Preparação do ambiente"
   ]
  },
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   "execution_count": 2,
   "id": "8aa7c21e-3cb8-4208-89bf-42a83bb3e730",
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   "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": 6,
   "id": "d24871e1-6a9a-4b38-807b-56591edd736d",
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    "tags": []
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   "source": [
    "import sympy as sym\n",
    "import numpy as np\n",
    "from sympy.plotting import PlotGrid\n",
    "from manim import *\n",
    "config.media_width = \"100%\"\n",
    "config.verbosity = \"WARNING\""
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d77d7ca7-f615-42db-8b45-61fe24bdabe5",
   "metadata": {},
   "source": [
    "## Definição do problema"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eec34521-570a-400c-b770-9482e867b879",
   "metadata": {},
   "source": [
    "Aqui, vamos encontrar uma expressão para $y(t)$ que só dependa de $n$, dada uma equação de diferenças linear e invariante no tempo:\n",
    "\n",
    "$$\\frac{d^Ny}{dt^N} + a_{N-1}\\frac{d^{N-1}y}{dt^{N-1}}+...+a_0y(t) = b_M\\frac{d^Mx}{dt^M} +...+b_2\\frac{d^2x}{dt^2}+b_1\\frac{dx}{dt} + b_0x(t) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca5a0654-9dfc-47aa-9184-92e5c838d4b7",
   "metadata": {},
   "source": [
    "## Composição da solução"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "A solução desta equação é formada por duas partes:\n",
    "\n",
    "$$y(t) = y_{zi}(t) + y_{zs}(t)$$\n",
    "\n",
    "em que $y_{zi}(t)$ é a parte de $y$ devida às condições iniciais do sinal de saída (os valores de $y(0), \\frac{dy}{dt}(0), ..., \\frac{d^{N-1}y}{dt^{N-1}}(0)$). Para o cálculo dessa componente, considera-se que a entrada $x(t)$ e todas as suas derivadas são nulas ($zi$ é de *zero input*, **entrada nula** em inglês). \n",
    "\n",
    "A outra componente, $y_{zs}(t)$, é devido à entrada $x(t)$. Para o cálculo dessa componente consideramos que os valores de $y(0), \\frac{dy}{dt}(0), ..., \\frac{d^{N-1}y}{dt^{N-1}}(0)$ são todos iguais a 0 ($zs$ é de *zero state*, **estado nulo** em inglês, às vezes também chamado de **repouso**)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bec022e9-0ff7-45ed-9cd0-8fc5d436a311",
   "metadata": {},
   "source": [
    "Aqui, vamos entender como obter a reposta de estado nulo para sistemas contínuos lineares e invariantes no tempo. A resposta devido às condições ininciais já foi abordada."
   ]
  },
  {
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   "source": [
    "## Integral de convolução\n",
    "\n",
    "Aqui, vamos desenvolver um método para encontrar a saída $y(t)$ de um sistema contínuo linear e invariante no tempo para uma entrada $x(t)$ qualquer.\n",
    "\n",
    "<img src=Imagens/contsis.png width=\"400\">"
   ]
  },
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   "cell_type": "markdown",
   "id": "e5885de7-b3aa-498c-976a-4cee43541b12",
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   "source": [
    "Vamos considerar que conhecemos a  resposta ao impulso do sistema.\n",
    "\n",
    "<img src=Imagens/diagimpulsocont.png width=\"300\">"
   ]
  },
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   "cell_type": "markdown",
   "id": "51f78208-f2dd-41e3-b227-5ee2545737a8",
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   "source": [
    "Como o sistema é linear e invariante no tempo se usarmos como entrada o impulso de Dirac multiplicado por um valor real e atrasado em alguma quantidade de tempo ($\\alpha\\delta(t-T)$), a saída será $\\alpha h(t-T)$ (multiplicada pelo mesmo valor e atrsada da mesma quantidade de tempo).\n",
    "\n",
    "<img src=Imagens/diagimpulsodelaycont.png width=\"400\">"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32bda570-32ab-4aed-898c-77445c381440",
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   "source": [
    "Em sinais de tempo contínuo, pelo terorema da amostragem, podemos escrever o valor de $x(t)$ em algum instante específico como:\n",
    "\n",
    "$$x(\\tau) = x(\\tau)\\delta(t-\\tau)$$\n",
    "\n",
    "Então ao usar $x(\\tau)$ como entrada, a saída é $x(\\tau)h(t-\\tau)$\n",
    "\n",
    "<img src=Imagens/diagimpulsoamostracont.png width=\"400\">"
   ]
  },
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   "source": [
    "O sinal $x(t)$ todo pode ser escrito como a soma (integral, já que é um sinal contínuo) dos valores em cada instante. Então podemos reescrever o sinal $x(t)$ como:\n",
    "\n",
    "$$x(t) = \\int_{-\\infty}^\\infty x(\\tau)\\delta(t-\\tau)\\,d\\tau$$\n",
    "\n",
    "Ou seja, o sinal $x(t)$ é a integral de impulsos deslocados no tempo. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64cb7906-3f2d-4201-98d1-6792cc0b3f9f",
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   "source": [
    "Então, considerando o exposto acima e o princípio da superposição que sistemas lineares e invariantes no tempo obedecem, temos que a saída do sistema será:\n",
    "\n",
    "\n",
    "$$y(t) = \\int_{-\\infty}^\\infty x(\\tau)h(t-\\tau)\\,d\\tau $$\n",
    "\n",
    "Esta relação está representada na imagem abaixo:\n",
    "\n",
    "<img src=Imagens/diagconvcont.png width=\"500\">\n",
    "\n",
    "\n",
    "Esta expressão é central no desenvolvimento da teoria de sistemas e sinais de tempo contínuo e recebe o nome de **integral de convolução**."
   ]
  },
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   "cell_type": "markdown",
   "id": "33799afc-7d0c-41a6-a13c-3f9bb37ecd78",
   "metadata": {},
   "source": [
    "A integral de convolução é uma operação entre dois sinais (no caso acima $x(t)$ e $h(t)$) e é representada pelo operador *.\n",
    "\n",
    "$$y(t) = \\int_{-\\infty}^\\infty x(\\tau)h(t-\\tau)\\,d\\tau = x(t)*h(t)$$ \n",
    "\n",
    "Perceba que o símbolo * é apenas uma forma reduzida de representar a integral de convolução (nunca se deve confundir com a simples multiplicação dos valores de $x$ e $h$ no instante $t$)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9c9b2fd-4282-40f0-b881-cac006151900",
   "metadata": {},
   "source": [
    "## Forma de calcular a integral de convolução\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d16f828-8840-4fde-b9a5-5accb0f133da",
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   "source": [
    "### Método gráfico\n",
    "\n",
    "Esse método é muito utilizado para saber a forma do resultado da convolução. É bom para entender o que representa a integral de convolução. Vamos voltar para a expressão da convolução:\n",
    "\n",
    "$$y(t) = x(t)*h(t) = \\int_{-\\infty}^\\infty x(\\tau)h(t-\\tau)\\,d\\tau$$ \n",
    "\n",
    "Podemos enxergar os sinais como sendo função de $\\tau$ ao invés de $t$. Então, para $t = 0$, temos os dois sinais a ser considerado: $x(t)$ e $h(-t)$ (a resposta ao impulso espelhada).\n",
    "\n",
    "---\n",
    "\n",
    "#### Exemplo 1\n",
    "\n",
    "Vamos fazer a convolução entre \n",
    "\n",
    "$$x(t) = 2u(t) - 2u(t-1)$$\n",
    "\n",
    "e\n",
    "\n",
    "$$h(t) = tu(t) - tu(t-2)$$"
   ]
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   "source": [
    "import sympy as sym\n",
    "import numpy as np\n",
    "from manim import *\n",
    "from sympy.plotting import PlotGrid\n",
    "config.media_width = \"100%\"\n",
    "config.verbosity = \"WARNING\""
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      "                                                                                                            \r"
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   "source": [
    "%%manim -qm IntegralConvolution\n",
    "\n",
    "def signal_1(t):\n",
    "    if 1 >= t >= 0:\n",
    "        return 2\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "def signal_2(t):\n",
    "    if 2 >= t>= 0:\n",
    "        return t\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "class IntegralConvolution(Scene):\n",
    "    def construct(self):\n",
    "        \n",
    "        axes1 = Axes(\n",
    "            x_range=[-5, 5, 1],\n",
    "            y_range=[0, 2, 1],\n",
    "            x_length=10,\n",
    "            y_length=1,\n",
    "            axis_config={\n",
    "                \"color\": WHITE,\n",
    "                \"stroke_width\": 2,\n",
    "                \"include_tip\": False,\n",
    "            },\n",
    "            x_axis_config={\n",
    "                \"numbers_to_include\": np.arange(-5, 6)\n",
    "            },\n",
    "            y_axis_config={\n",
    "                \"numbers_to_include\": np.arange(0, 2, 1),\n",
    "            },\n",
    "        ).move_to(3.1*UP)\n",
    "        self.play(Create(axes1))\n",
    "        \n",
    "        \n",
    "        line_1 = axes1.plot(lambda x: signal_1(x), color=BLUE, \n",
    "                            x_range=[-5,5, 0.01], use_smoothing=False)\n",
    "        tex1 = MathTex('x(\\\\tau)').move_to((-3.7,3.7,0))\n",
    "        self.play(Create(line_1), Create(tex1))\n",
    "        \n",
    "        axes2 = Axes(\n",
    "            x_range=[-5, 5, 1],\n",
    "            y_range=[0, 2, 1],\n",
    "            x_length=10,\n",
    "            y_length=1,\n",
    "            axis_config={\n",
    "                \"color\": WHITE,\n",
    "                \"stroke_width\": 2,\n",
    "                \"include_tip\": False,\n",
    "            },\n",
    "            x_axis_config={\n",
    "                \"numbers_to_include\": np.arange(-5, 6)\n",
    "            },\n",
    "            y_axis_config={\n",
    "                \"numbers_to_include\": np.arange(0, 2, 1),\n",
    "            },\n",
    "        ).move_to(1.4*UP)\n",
    "        self.play(Create(axes2))\n",
    "        line_2 = axes2.plot(lambda x: signal_2(x), color=RED, \n",
    "                            x_range=[-5, 5, 0.01], use_smoothing=False)\n",
    "        tex2 = MathTex('h(\\\\tau)').move_to((-3.7,1.7,0))\n",
    "        self.play(Create(line_2), Create(tex2))\n",
    "        line_3 = axes2.plot(lambda x: signal_2(-x), color=RED, \n",
    "                            x_range=[-5, 5, 0.01], use_smoothing=False)\n",
    "        tex3 = MathTex('h(\\\\tau)').move_to((-3.7,1.7,0))\n",
    "        self.play(ReplacementTransform(line_2,line_3), \n",
    "                  ReplacementTransform(tex2, tex3))        \n",
    "        axes3 = Axes(     \n",
    "            x_range=[-5, 5, 1],\n",
    "            y_range=[0, 2, 2],\n",
    "            x_length=10,\n",
    "            y_length=1,\n",
    "            axis_config={\n",
    "                \"color\": WHITE,\n",
    "                \"stroke_width\": 2,\n",
    "                \"include_tip\": False,\n",
    "            },\n",
    "            x_axis_config={\n",
    "                \"numbers_to_include\": np.arange(-5, 6)\n",
    "            },\n",
    "            y_axis_config={\n",
    "                \"numbers_to_include\": np.arange(0, 2, 1),\n",
    "            },\n",
    "        ).move_to(0.5*DOWN)\n",
    "        self.play(Create(axes3))\n",
    "        \n",
    "        axes4 = Axes(\n",
    "            x_range=[-5, 5, 1],\n",
    "            y_range=[0, 2, 1],\n",
    "            x_length=10,\n",
    "            y_length=1,\n",
    "            axis_config={\n",
    "                \"color\": WHITE,\n",
    "                \"stroke_width\": 2,\n",
    "                \"include_tip\": False,\n",
    "            },\n",
    "            x_axis_config={\n",
    "                \"numbers_to_include\": np.arange(-5, 6)\n",
    "            },\n",
    "            y_axis_config={\n",
    "                \"numbers_to_include\": np.arange(0, 2, 1),\n",
    "            },\n",
    "        ).move_to(2.4*DOWN)\n",
    "        self.play(Create(axes4))\n",
    "        \n",
    "        dx = 0.01\n",
    "        x = np.arange(-5,5, dx)\n",
    "        sig1 = np.zeros_like(x)\n",
    "        sig2 = np.zeros_like(x)\n",
    "        line_4 = axes3.plot_line_graph(x, sig1*sig2, \n",
    "                                       add_vertex_dots=False, \n",
    "                                       line_color=GREEN)\n",
    "        \n",
    "        tex4 = MathTex('x(\\\\tau)h(t-\\\\tau)').move_to((-3.7,-0.3,0))\n",
    "        tex5 = MathTex('\\int_{-\\infty}^\\infty x(\\\\tau)h(t-\\\\tau)\\,d\\\\tau').move_to((-3.7,-2.3,0))\n",
    "        tex6 = MathTex('h(t-\\\\tau)').move_to((-3.7,1.7,0))\n",
    "        \n",
    "      \n",
    "        # area = Polygon([-5,0,0], color=GREEN)\n",
    "        self.play(Create(line_4), Create(tex4), \n",
    "                  Create(tex5), ReplacementTransform(tex3, tex6),\n",
    "                  )\n",
    "        sigconv = []\n",
    "        \n",
    "        \n",
    "        \n",
    "        for t in np.arange(-5, 5, 0.1):\n",
    "            \n",
    "            \n",
    "           \n",
    "            for i in range(len(x)):\n",
    "                sig1[i] = signal_1(x[i])\n",
    "                sig2[i] = signal_2(t-x[i])\n",
    "            conv = np.sum(dx*(sig1*sig2))\n",
    "            sigconv.append(conv)\n",
    "            line_5 = axes3.plot_line_graph(x, sig1*sig2, \n",
    "                                           add_vertex_dots=False, \n",
    "                                           line_color=GREEN)\n",
    "            listpoints_area = [[-5,0,0]]\n",
    "            for j in range(len(sig1)):\n",
    "                listpoints_area.append([j*dx-5, sig1[j]*sig2[j], 0])\n",
    "            listpoints_area.append([5, 0,0])\n",
    "            \n",
    "            #area1 = Polygon(*listpoints_area, color=GREEN)\n",
    "            \n",
    "            line_6 = axes4.plot_line_graph(x[:10*len(sigconv):10], \n",
    "                                           np.array(sigconv), \n",
    "                                           add_vertex_dots=False, \n",
    "                                           line_color=GOLD)\n",
    "            self.play(line_3.animate.move_to(axes2.c2p(*(t, 1,0))),\n",
    "                      Transform(line_4, line_5),\n",
    "                      Create(line_6),\n",
    "                      run_time=0.3)\n",
    "            \n",
    "          \n",
    "            \n",
    "        self.wait(5)"
   ]
  },
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    "\n",
    "---\n",
    "\n",
    "#### Resumo dos passos\n",
    "\n",
    "1 - substituir $t$ por $\\tau$ e fazer o gráfico de $x(\\tau)$ e $h(-\\tau)$.\n",
    "\n",
    "\n",
    "2 - Deslocar $h[-\\tau]$ para a direita em algumas unidades de tempo para gerar $h(t-\\tau)$ até um ponto em que $h(t-\\tau)$ não se sobreponha a $x(\\tau)$. Multiplique os dois sinais e calcule a integral (área) do sinal resultante para encontrar o valor de $y(t)$. \n",
    "\n",
    "3 - Continue deslocando $h(-\\tau)$ para a direita, multiplicando o sinal $x(t)$ com a resposta ao impulso deslocada e calculando a sua integral  para encontrar os valores de $y(t)$. Continue fazendo isso até não ter mais sobreposição entre os sinais."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca53782f",
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   "source": [
    "### Pelo cálculo da integral\n",
    "\n",
    "Muitas vezes, é mais fácil calcular a integral de convolução diretamente, sem recorrer ao método gráfico. \n",
    "\n",
    "---\n",
    "\n",
    "#### Exemplo 2\n",
    "\n",
    "$$x(t) = e^{-bt}u(t)$$ \n",
    "\n",
    "e\n",
    "\n",
    "$$h(t) = e^{-at}u(t)$$\n",
    "\n",
    "A integral de convolução entre esses dois sinais é:\n",
    "\n",
    "$$x(t)*h(t) = \\int_{-\\infty}^\\infty e^{-b\\tau}u(\\tau)e^{-a(t-\\tau)}u(t-\\tau)\\,d\\tau=\\int_{0}^t e^{-b\\tau}e^{-a(t-\\tau)}\\,d\\tau = e^{at}\\int_{0}^t e^{-b\\tau}e^{a\\tau)}\\,d\\tau $$\n",
    "\n",
    "$$x(t)*h(t) = e^{at}\\int_{0}^t e^{(a-b)\\tau}\\,d\\tau = \\left. e^{at}\\left( \\frac{e^{(a-b)\\tau}}{a-b}\\right)\\right|_{\\tau=0}^t\\frac{e^{-bt}-e^{-at}}{a-b}$$\n",
    "\n",
    "Para $a=2$ e $b=5$, o gráfico dessa convolução é:"
   ]
  },
  {
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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 0x7f79f7c088b0>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sym.symbols('t')\n",
    "a = 2\n",
    "b = 5\n",
    "y = (sym.exp(-b*t) - sym.exp(-a*t))/(a-b)\n",
    "sym.plot(y, (t, 0, 6))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d197a7b4",
   "metadata": {},
   "source": [
    "É claro que poderíamos ter feito essa convolução com o método gráfico também."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "4e40e3ac",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "                                                                                \r"
     ]
    },
    {
     "data": {
      "text/html": [
       "<video src=\"media/jupyter/IntegralConvolution1@2023-06-22@10-41-42.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 IntegralConvolution1\n",
    "\n",
    "def signal_1(t):\n",
    "    if t >= 0:\n",
    "        return np.exp(-2*t)\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "def signal_2(t):\n",
    "    if t>= 0:\n",
    "        return np.exp(-5*t)\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "class IntegralConvolution1(Scene):\n",
    "    def construct(self):\n",
    "        \n",
    "        axes1 = Axes(\n",
    "            x_range=[-5, 5, 1],\n",
    "            y_range=[0, 2, 1],\n",
    "            x_length=10,\n",
    "            y_length=1,\n",
    "            axis_config={\n",
    "                \"color\": WHITE,\n",
    "                \"stroke_width\": 2,\n",
    "                \"include_tip\": False,\n",
    "            },\n",
    "            x_axis_config={\n",
    "                \"numbers_to_include\": np.arange(-5, 6)\n",
    "            },\n",
    "            y_axis_config={\n",
    "                \"numbers_to_include\": np.arange(0, 2, 1),\n",
    "            },\n",
    "        ).move_to(3.1*UP)\n",
    "        self.play(Create(axes1))\n",
    "        \n",
    "        \n",
    "        line_1 = axes1.plot(lambda x: signal_1(x), color=BLUE, \n",
    "                            x_range=[-5,5, 0.01], use_smoothing=False)\n",
    "        tex1 = MathTex('x(\\\\tau)').move_to((-3.7,3.7,0))\n",
    "        self.play(Create(line_1), Create(tex1))\n",
    "        \n",
    "        axes2 = Axes(\n",
    "            x_range=[-5, 5, 1],\n",
    "            y_range=[0, 2, 1],\n",
    "            x_length=10,\n",
    "            y_length=1,\n",
    "            axis_config={\n",
    "                \"color\": WHITE,\n",
    "                \"stroke_width\": 2,\n",
    "                \"include_tip\": False,\n",
    "            },\n",
    "            x_axis_config={\n",
    "                \"numbers_to_include\": np.arange(-5, 6)\n",
    "            },\n",
    "            y_axis_config={\n",
    "                \"numbers_to_include\": np.arange(0, 2, 1),\n",
    "            },\n",
    "        ).move_to(1.4*UP)\n",
    "        self.play(Create(axes2))\n",
    "        line_2 = axes2.plot(lambda x: signal_2(x), color=RED, \n",
    "                            x_range=[-5, 5, 0.01], use_smoothing=False)\n",
    "        tex2 = MathTex('h(\\\\tau)').move_to((-3.7,1.7,0))\n",
    "        self.play(Create(line_2), Create(tex2))\n",
    "        line_3 = axes2.plot(lambda x: signal_2(-x), color=RED, \n",
    "                            x_range=[-5, 5, 0.01], use_smoothing=False)\n",
    "        tex3 = MathTex('h(\\\\tau)').move_to((-3.7,1.7,0))\n",
    "        self.play(ReplacementTransform(line_2,line_3), \n",
    "                  ReplacementTransform(tex2, tex3))        \n",
    "        axes3 = Axes(     \n",
    "            x_range=[-5, 5, 1],\n",
    "            y_range=[0, 2, 2],\n",
    "            x_length=10,\n",
    "            y_length=1,\n",
    "            axis_config={\n",
    "                \"color\": WHITE,\n",
    "                \"stroke_width\": 2,\n",
    "                \"include_tip\": False,\n",
    "            },\n",
    "            x_axis_config={\n",
    "                \"numbers_to_include\": np.arange(-5, 6)\n",
    "            },\n",
    "            y_axis_config={\n",
    "                \"numbers_to_include\": np.arange(0, 2, 1),\n",
    "            },\n",
    "        ).move_to(0.5*DOWN)\n",
    "        self.play(Create(axes3))\n",
    "        \n",
    "        axes4 = Axes(\n",
    "            x_range=[-5, 5, 1],\n",
    "            y_range=[0, 0.2, 1],\n",
    "            x_length=10,\n",
    "            y_length=1,\n",
    "            axis_config={\n",
    "                \"color\": WHITE,\n",
    "                \"stroke_width\": 2,\n",
    "                \"include_tip\": False,\n",
    "            },\n",
    "            x_axis_config={\n",
    "                \"numbers_to_include\": np.arange(-5, 6)\n",
    "            },\n",
    "            y_axis_config={\n",
    "                \"numbers_to_include\": np.arange(0, .3, 0.1),\n",
    "            },\n",
    "        ).move_to(2.4*DOWN)\n",
    "        self.play(Create(axes4))\n",
    "        \n",
    "        dx = 0.01\n",
    "        x = np.arange(-5,5, dx)\n",
    "        sig1 = np.zeros_like(x)\n",
    "        sig2 = np.zeros_like(x)\n",
    "        line_4 = axes3.plot_line_graph(x, sig1*sig2, \n",
    "                                       add_vertex_dots=False, \n",
    "                                       line_color=GREEN)\n",
    "        \n",
    "        tex4 = MathTex('x(\\\\tau)h(t-\\\\tau)').move_to((-3.7,-0.3,0))\n",
    "        tex5 = MathTex('\\int_{-\\infty}^\\infty x(\\\\tau)h(t-\\\\tau)\\,d\\\\tau').move_to((-3.7,-2.3,0))\n",
    "        tex6 = MathTex('h(t-\\\\tau)').move_to((-3.7,1.7,0))\n",
    "        \n",
    "      \n",
    "        # area = Polygon([-5,0,0], color=GREEN)\n",
    "        self.play(Create(line_4), Create(tex4), \n",
    "                  Create(tex5), ReplacementTransform(tex3, tex6),\n",
    "                  )\n",
    "        sigconv = []\n",
    "        \n",
    "        \n",
    "        \n",
    "        for t in np.arange(-5, 5, 0.1):           \n",
    "           \n",
    "            for i in range(len(x)):\n",
    "                sig1[i] = signal_1(x[i])\n",
    "                sig2[i] = signal_2(t-x[i])\n",
    "            conv = np.sum(dx*(sig1*sig2))\n",
    "            sigconv.append(conv)\n",
    "            line_5 = axes3.plot_line_graph(x, sig1*sig2, \n",
    "                                           add_vertex_dots=False, \n",
    "                                           line_color=GREEN)\n",
    "            \n",
    "            \n",
    "            #area1 = Polygon(*listpoints_area, color=GREEN)\n",
    "            \n",
    "            line_6 = axes4.plot_line_graph(x[:10*len(sigconv):10], \n",
    "                                           np.array(sigconv), \n",
    "                                           add_vertex_dots=False, \n",
    "                                           line_color=GOLD)\n",
    "            self.play(line_3.animate.move_to(axes2.c2p(*(t, 0.5,0))),\n",
    "                      Transform(line_4, line_5),\n",
    "                      Create(line_6),\n",
    "                      run_time=0.3)\n",
    "            \n",
    "          \n",
    "            \n",
    "        self.wait(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d271439-fa05-41d9-b2f8-f4f1e0314e92",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "#### Exemplo 3\n",
    "\n",
    "Vamos considerar um sistema com a seguinte resposta ao impulso:\n",
    "\n",
    "$h(t) = e^{-3t}u(t)$\n",
    "\n",
    "Se esse sistema tiver como entrada:\n",
    "\n",
    "$x(t) = u(t)$\n",
    "\n",
    "Então, a saída devido a essa entrada é:\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty u(\\tau)e^{-3(t-\\tau)}u(t-\\tau)\\,d\\tau = \\int_{0}^t e^{-3(t-\\tau)}\\,d\\tau = e^{-3t}\\int_{0}^t e^{3\\tau}\\,d\\tau = \\frac{e^{-3t}}{3}\\left.e^{3\\tau}\\right|_{\\tau=0}^t u(t) = \\frac{e^{-3t}}{3}(e^{3t}-1) = \\frac{1-e^{-3t}}{3}u(t)$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "f7e0520e-212a-42ad-bca5-6769461f12b2",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = sym.symbols('t')\n",
    "y = (1-sym.exp(-3*t))/3*sym.Heaviside(t)\n",
    "sym.plot(y, (t, -1, 6));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "74a1dcd5-bd0e-4a4a-9870-9619c96164d3",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df444225-a9c7-4e2c-83a6-7cd97ced1641",
   "metadata": {},
   "source": [
    "## Caracterização de um SLIT pela resposta ao impulso\n",
    "\n",
    "A principal consequência da integral de convolução é que conseguimos caracterizar completamente um sistema linear e invariante no tempo de contínuo pela sua resposta ao impulso. Perceba que não é necessário conhecer a equação de diferenças nem o diagrama de blocos do sistema se tivermos a resposta ao impulso. Portanto, **a resposta ao impulso caracteriza completamente o comportamento de um sistema linear e invariante no tempo**.\n",
    "\n",
    "Por isso, é bastante comum no diagrama de blocos representar o sistema pela sua respsota ao impulso.\n",
    "\n",
    "\n",
    "<img src=Imagens/impulsoconvcont.png width=\"500\">"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db531caf-f544-4716-80b2-91f5f8b4bb63",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "## Propriedades da convolução\n",
    "\n",
    "### Comutativa\n",
    "\n",
    "Uma das propriedades mais importantes, principalmente do ponto de vista prático, já que pode facilitar a computação da conolução.\n",
    "\n",
    "> $x(t)*h(t) = h(t)*x(t)$\n",
    "\n",
    "**Prova**: \n",
    "\n",
    "Aqui é feita uma mudança de variável ($\\upsilon = t-\\tau$), e a prova sai facilmente. \n",
    "\n",
    "$$ x(t)*h(t) =  \\int_{-\\infty}^\\infty x(\\tau)h(t-\\tau)\\,d\\tau = \\int_{-\\infty}^\\infty x(t-\\upsilon)h(\\upsilon)\\,d\\upsilon = h(t)*x(t)$$\n",
    "\n",
    "\n",
    "### Distributiva\n",
    "\n",
    "> $x(t)*h_1(t)+x(t)*h_2(t) = x(t)*(h_1(t)+ h_2(t))$\n",
    "\n",
    "A prova dessa propriedade fica para o Exercício 1.  \n",
    "\n",
    "Com esta propriedade, podemos interpretar dois sistemas ligados em paralelo, recebendo a mesma entrada, como sendo apenas um sistema que tem a resposta ao impulso igual a soma das respostas ao impulso dos dois sistemas ligados em paraleo.\n",
    "\n",
    "<img src=Imagens/distributivaconvcont.png width=\"600\">\n",
    "\n",
    "\n",
    "Uma outra consequência dessa propriedade é que podemos dividir a resposta ao impulso de um sistema em várias partes se isso facilitar o cálculo da somatória de convolução.\n",
    "\n",
    "### Associativa\n",
    "\n",
    "> $x(t)*(h_1(t)*h_2(t)) = (x(t)*h_1(t))*h_2(t)$\n",
    "\n",
    "A prova dessa propriedade fica para o Exercício 2. \n",
    "\n",
    "A principal consequência dessa propriedade é que se tivermos dois sistemas em série, podemos considerá-los como um sistema com a resposta ao impulso sendo a convolução das respostas ao impulso dos dois sistemas.\n",
    "\n",
    "<img src=Imagens/associativaconvcont.png width=\"500\">\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cfbb8de3-484c-4fe5-a0c6-e25fda84a999",
   "metadata": {},
   "source": [
    "## Exercícios"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "375a538f-2897-4e22-951c-d98b8531f726",
   "metadata": {},
   "source": [
    "**1)** Prove a propriedade distributiva da integral de convolução."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ce8f6a4-a4b2-4c9d-b034-479c1bf82a66",
   "metadata": {},
   "source": [
    "**2)** Prove a propriedade associativa da integral de convolução"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e46c7139-1343-4a03-b792-3c961b01c5a9",
   "metadata": {},
   "source": [
    "**3)** Seja o sinal $h(t)$:\n",
    "\n",
    "$$h(t) = e^{3t}u(-t+1) + e^{-3t}u(t-3)$$\n",
    "\n",
    "Determine os valores de $A$ e $B$ de forma que:\n",
    "\n",
    "$$h(t-\\tau) = \\left\\{\\begin{array}{cc}e^{-3(t-\\tau)}&,\\tau<A\\\\0&,A\\leq\\tau<B \\\\e^{3(t-\\tau)}&,\\tau\\geq B \\end{array} \\right.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f1f1ebc-2a4c-4e47-814b-6c21b089a398",
   "metadata": {},
   "source": [
    "**4)** Determine:\n",
    "\n",
    "a) $u(t)*u(t)$\n",
    "\n",
    "b) $e^{-3t}u(t)*u(t)$\n",
    "\n",
    "c) $tu(t)*u(t)$\n",
    "\n",
    "d) $\\mathrm{sen}(t)u(t)*u(t)$\n",
    "\n",
    "e) $\\cos(t)u(t)*u(t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52df0ca4-df52-4295-b556-a9548dbf09b1",
   "metadata": {},
   "source": [
    "**5)** Determine a resposta de um sistema linear e invariante no tempo (estado nulo) às  entradas abaixo, considerando que esse sistema tem a seguinte resposta ao impulso:\n",
    "\n",
    "$h(t) = e^{-3t}u(t)$\n",
    "\n",
    "a) $x_1(t) = u(t)$\n",
    "\n",
    "b) $x_2(t) = e^{-2t}u(t)$\n",
    "\n",
    "c) $x_3(t) = \\cos(2t)u(t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcaf6126-36fd-4f2d-8ff2-3d9e56a4287f",
   "metadata": {},
   "source": [
    "**6)** Calcule a resposta do sistema que tem a resposta ao impulso $h(t)$ mostrada no gráfico abaixo quando é usada a entrada $x(t)$, também mostrada abaixo. Esse sistema é causal?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "aafbc3c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.PlotGrid at 0x7f79ef7155a0>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sym.symbols('t')\n",
    "\n",
    "x = sym.Piecewise((1, (t >= 0)),\n",
    "                  (0, True))\n",
    "p1 = sym.plot(x, (t, -3, 3), show=False, title='$x(t)$')\n",
    "h = sym.Piecewise((t+1, (t >= -1) & (t < 0)),\n",
    "                  (0, True))\n",
    "p2 = sym.plot(h, (t, -3,3), show=False, title='$h(t)$')\n",
    "PlotGrid(1, 2, p1, p2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "604f175e-c0be-4ce3-b286-6c8e0cbf366a",
   "metadata": {},
   "source": [
    "**7)** 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 capacitor $y(t)=V_c$. Pede-se:\n",
    "\n",
    "a) A equação diferencial que descreve o comportamento desse sistema (lembre-se que $I=C\\frac{dV_c}{dt}$ e $I = \\frac{V_r}{R}$ e $V_{in}=V_R+V_C$).\n",
    "\n",
    "b) A resposta ao impulso desse sistema.\n",
    "\n",
    "c) Determine a resposta de entrada nula, e esboce seu gráfico, considerando que $V_c(0) = 1$ V.\n",
    "\n",
    "d) Encontre o sinal de saída, levando em conta a sua condição inicial, para o sinal de entrada $x(t) = u(t)$ V."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71f7d111-b25c-492e-ace3-da5959f5fed0",
   "metadata": {
    "tags": []
   },
   "source": [
    "**8)** Qual o comportamento esperado (descreva em palavras, justificando) de um sistema que tenha como resposta ao impulso as seguintes funções:\n",
    "\n",
    "a) $h_1(t) = u(t)$\n",
    "\n",
    "b) $h_2(t) = \\delta(t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9769381-7d55-422b-920e-d4d74066ec1a",
   "metadata": {},
   "source": [
    "**9)** Considerando que um sistema linear e invariante no tempo com resposta ao impulso $h(t)$ tem uma saída $y(t)$ quando é usada como entrada o sinal $x(t)$. Qual é a saída do sistema quando a entrada é $x_1(t) = \\frac{dx}{dt}$?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78442a40-a434-47d6-8c48-88a2ec319406",
   "metadata": {},
   "source": [
    "**10)** Encontre a solução de estado nulo para os sistemas descritos pelas seguintes equações diferenciais e entradas.\n",
    "\n",
    "a) $\\frac{dy}{dt} = -6y(t) + x(t)$, com $x(t)=u(t)$\n",
    "\n",
    "b) $\\frac{dy}{dt} = 4y(t)  + x(t)$, com $x(t) = e^{-3t}u(t)$\n",
    "\n",
    "c) $\\frac{d^2y}{dt^2} = -\\frac{dy}{dt} + \\frac{dx}{dt} + x(t)$, com $x(t) = 25u(t)$\n",
    "\n",
    "d) $\\frac{d^2y}{dt^2} = -4y(t) + x(t)$, com $x(t) = \\cos(2t)u(t)$\n",
    "\n",
    "e) $\\frac{d^2y}{dt^2} = -4\\frac{dy}{dt}-13y(t) + 4\\frac{dx}{dt} + 2x(t)$, com $x(t) = e^{-t}u(t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "974e97a2-d4c5-45b8-9afd-0093def37fb2",
   "metadata": {},
   "source": [
    "## Referências\n",
    "\n",
    "- Ulaby, FT e Yagle, AE, Signals & Systems: Theory and Applications, 2018, [ss2.eecs.umich.edu](ss2.eecs.umich.edu)\n",
    "- Oppenheim, M.  e Willsky, S., Sistemas e Sinais, 2010\n",
    "\n",
    "- Lathi, BP, Sinais e Sistemas Lineares, 2007"
   ]
  },
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   "source": [
    "## Solução dos exercícios"
   ]
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    "**1)** Resolvido por Juliana Soares da Silva de Oliveira\n",
    "\n",
    "Prove a propriedade distributiva da integral de convolução.\n",
    "\n",
    "Partindo de:\n",
    "\n",
    "$x(t)*h_1(t)+x(t)*h_2(t) = x(t)*(h_1(t)+ h_2(t))$\n",
    "\n",
    "Aplicando a definição na parte esquerda da equação:\n",
    "\n",
    "$\\int_{-\\infty}^\\infty x(\\tau)h_1(t-\\tau)\\,d\\tau + \\int_{-\\infty}^\\infty x(\\tau)h_2(t-\\tau)\\,d\\tau$\n",
    "\n",
    "$\\int_{-\\infty}^\\infty (x(\\tau)h_1(t-\\tau) + x(\\tau)h_2(t-\\tau))\\,d\\tau$\n",
    "\n",
    "$\\int_{-\\infty}^\\infty x(\\tau)(h_1(t-\\tau) + h_2(t-\\tau))\\,d\\tau$\n",
    "\n",
    "vamos definir uma função $h_{3}(t-\\tau) = h_1(t-\\tau) + h_2(t-\\tau)$, sendo assim:\n",
    "\n",
    "$\\int_{-\\infty}^\\infty x(\\tau)h_{3}(t-\\tau)\\,d\\tau = x(t)*h_3(t) = x(t)*(h_1(t)+h_2(t))$"
   ]
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   "source": [
    "**4)** Resolvido por Juliana Soares da Silva de Oliveira\n",
    "\n",
    "Determine:\n",
    "\n",
    "a) $u(t)*u(t)$\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty u(\\tau)u(t-\\tau)\\,d\\tau = \\int_{0}^t 1\\,d\\tau = \\tau|_{\\tau=0}^t = tu(t)$\n",
    "\n",
    "\n",
    "b) $e^{-3t}u(t)*u(t)$\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty e^{-3\\tau}u(\\tau)u(t-\\tau)\\,d\\tau = \\int_{0}^t e^{-3\\tau}\\,d\\tau = \\frac{e^{-3\\tau}}{-3}|_{\\tau=0}^t = \\frac{e^{-3t}-1}{-3}u(t)$\n",
    "\n",
    "\n",
    "c) $tu(t)*u(t)$\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty \\tau u(\\tau)u(t-\\tau)\\,d\\tau = \\int_{0}^t \\tau\\,d\\tau = \\frac{\\tau^2}{2}|_{\\tau=0}^t = \\frac{t^2}{2}u(t)$\n",
    "\n",
    "\n",
    "d) $\\mathrm{sen}(t)u(t)*u(t)$\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty \\mathrm{sen}(\\tau)u(\\tau)u(t-\\tau)\\,d\\tau = \\int_{0}^t \\mathrm{sen}(\\tau)\\,d\\tau = -\\cos(\\tau)|_{\\tau=0}^t = (-\\cos(t)+1)u(t)$\n",
    "\n",
    "e) $\\cos(t)u(t)*u(t)$\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty \\cos(\\tau)u(\\tau)u(t-\\tau)\\,d\\tau = \\int_{0}^t \\cos(\\tau)\\,d\\tau = \\mathrm{sen}(\\tau)|_{\\tau=0}^t = \\mathrm{sen}(t)u(t)$"
   ]
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   "id": "e65a1acc-e4c0-4a28-a019-dad49333cc4a",
   "metadata": {},
   "source": [
    "**5)** Resolvido por Juliana Soares da Silva de Oliveira\n",
    "\n",
    "Determine a resposta de um sistema linear e invariante no tempo (estado nulo) às  entradas abaixo, considerando que esse sistema tem a seguinte resposta ao impulso:\n",
    "\n",
    "$h(t) = e^{-3t}u(t)$\n",
    "\n",
    "a) $x_1(t) = u(t)$\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty u(\\tau)e^{-3(t-\\tau)}u(t-\\tau)\\,d\\tau = \\int_{0}^t e^{-3(t-\\tau)}\\,d\\tau = e^{-3t}\\int_{0}^t e^{3\\tau}\\,d\\tau = \\frac{e^{-3t}}{3}\\left.e^{3\\tau}\\right|_{\\tau=0}^t u(t) = \\frac{e^{-3t}}{3}(e^{3t}-1) = \\frac{1-e^{-3t}}{3}u(t)$\n",
    "\n",
    "\n",
    "b) $x_2(t) = e^{-2t}u(t)$\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty e^{-2\\tau}u(\\tau)e^{-3(t-\\tau)}u(t-\\tau)\\,d\\tau = \\int_{0}^t e^{-2\\tau}e^{-3(t-\\tau)}\\,d\\tau = e^{-3t}\\int_{0}^t e^{-2\\tau}e^{3\\tau}\\,d\\tau = e^{-3t}\\int_{0}^t e^{\\tau}\\,d\\tau = e^{-3t}e^{\\tau}|_{\\tau=0}^t u(t) = e^{-3t}(e^{t}-1) = (e^{-2t}-e^{-3t})u(t)$\n",
    "\n",
    "\n",
    "c) $x_3(t) = \\cos(2t)u(t)$\n",
    "\n",
    "$y(t) = \\int_{-\\infty}^\\infty \\cos(2\\tau)u(\\tau)e^{-3(t-\\tau)}u(t-\\tau)\\,d\\tau = \\int_{0}^t \\cos(2\\tau)e^{-3(t-\\tau)}\\,d\\tau = e^{-3t}\\int_{0}^t \\cos(2\\tau)e^{3\\tau}\\,d\\tau = e^{-3t}\\int_{0}^t \\frac{(e)^{j2\\tau}+(e)^{-j2\\tau}}{2}e^{3\\tau}\\,d\\tau = e^{-3t}\\int_{0}^t \\frac{(e)^{\\tau(3+j2)}+(e)^{\\tau(3-j2)}}{2}\\,d\\tau = \\frac{e^{-3t}}{2}(\\int_{0}^t (e)^{\\tau(3+j2)}\\,d\\tau + \\int_{0}^t (e)^{\\tau(3-j2)}\\,d\\tau) = $\n",
    "\n",
    "$y(t)=\\frac{e^{-3t}}{2}(\\frac{e^{\\tau(3+j2)}}{3+j2}+\\frac{e^{\\tau(3-j2)}}{3-j2})|_{\\tau=0}^t = \\frac{e^{-3t}}{2}(\\frac{e^{(3+j2)t}-1}{3+j2}+\\frac{e^{(3-j2)t}-1}{3-j2})u(t) = \\frac{1}{2}(\\frac{e^{(j2)t}-e^{-3t}}{3+j2}+\\frac{e^{(-j2)t}-e^{-3t}}{3-j2})u(t)=\\frac{1}{2}(\\frac{e^{(j2)t}-e^{-3t}}{\\sqrt{13}e^{j\\arctan(\\frac{2}{3})}}+\\frac{e^{(-j2)t}-e^{-3t}}{\\sqrt{13}e^{-j\\arctan(\\frac{2}{3})}})u(t)$\n",
    "\n",
    "$y(t) = \\frac{\\sqrt{13}}{13}\\cos(2t+\\arctan(\\frac{2}{3}))-\\frac{\\sqrt{13}e^{-3t}}{13}\\cos(\\arctan(\\frac{2}{3})u(t)$"
   ]
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   "execution_count": null,
   "id": "a5516d9a-730b-4448-afef-bc30c37da38e",
   "metadata": {},
   "outputs": [],
   "source": []
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