Exemplos_apostila_PDS.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Exemplos_apostila_PDS.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyOy0BgakvgsvegcP9EdXRzH",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/AlyssonBatista/Codigos-python/blob/main/Exemplos_apostila_PDS.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Capitulo 2\n",
        "Sinais e Sistemas de Tempo\n",
        "Discreto"
      ],
      "metadata": {
        "id": "FSc1og68wGnk"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "qTmcp5cAv9qf",
        "outputId": "1d3f6ccd-75ee-46b7-8cac-3e1c49d0cdc1"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "\n",
        "n=np.arange(-3,4)\n",
        "x=np.arange(-3,4)\n",
        "\n",
        "plt.style.use('Solarize_Light2')\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.title(\"x[n]\")\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Impulson initário δ(t)"
      ],
      "metadata": {
        "id": "vdFpN45x_Kt3"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def impseq(n0,n1,n2):\n",
        "    n = np.arange(n1,n2+1)\n",
        "    x = np.where(n-n0==0,1,0)\n",
        "    return [n,x]\n",
        "\n",
        "n = impseq(0,-5,5)\n",
        "x = impseq(0,-5,5)\n",
        "\n",
        "plt.stem(n[0],x[1],use_line_collection=True)\n",
        "plt.title(\"δ[n]\")\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "RLQozefz1RLu",
        "outputId": "31270d55-1032-4d6d-fccf-6a8ceb4a727e"
      },
      "execution_count": 54,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Degrau unitário"
      ],
      "metadata": {
        "id": "Eu3j1LDsU8zC"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def stepseq(n0,n1,n2):\n",
        "    if n0<n1 or n0>n2 or n1>n2:\n",
        "        raise ValueError(\"é necessário satisfazer n1 <= n0 <= n2\")\n",
        "    else:\n",
        "        n = np.arange(n1,n2+1)\n",
        "        x = np.where(n-n0 >= 0,1,0)\n",
        "    return [n,x]\n",
        "\n",
        "n = stepseq(0,-10,10)\n",
        "x = stepseq(0,-10,10)\n",
        "plt.stem(n[0],x[1],use_line_collection=True)\n",
        "plt.title('u[n]')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "wIWWc2pX9PK4",
        "outputId": "799cbe9d-9b7d-4d0a-f0c8-05a3e2ca0037"
      },
      "execution_count": 90,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Exercícios - exibir os gráficos dos seguintes sinais \n",
        "\n"
      ],
      "metadata": {
        "id": "WWOx40z9XUSQ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "### 1. impseq(6,1,11)"
      ],
      "metadata": {
        "id": "TA2Qayn2Zc78"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = impseq(6,1,11)\n",
        "x = impseq(6,1,11)\n",
        "\n",
        "plt.stem(n[0],x[1],use_line_collection=True)\n",
        "plt.title(\"δ[n-6]\")\n",
        "plt.show()\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "cHjgmNFzXgSi",
        "outputId": "d5a69b21-6463-4beb-e536-06be90461d6d"
      },
      "execution_count": 91,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 2. x[n] = 2 × δ[n + 2] - δ[n - 4], -5 <= n <= 5"
      ],
      "metadata": {
        "id": "UWRztfdOZjs3"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "x = 2 * impseq(-2,-5,5)[1] - impseq(4,-5,5)[1]\n",
        "n = np.arange(-5,5+1)\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.title(\"x[n] = 2 . δ[n + 2] - δ[n - 4]\")\n",
        "plt.show()\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "W_zZ89KzZuQA",
        "outputId": "3457fb0d-ad21-4393-ddaa-60bb7a20fa59"
      },
      "execution_count": 82,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 3.stepseq(6,1,11)"
      ],
      "metadata": {
        "id": "4bODZxvhdXmm"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n,x = stepseq(6,1,11)\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.title(\"u[n-6]\")\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "_80j2yyQdhcK",
        "outputId": "50ee9291-1530-4ae1-93ea-9da9b34f44aa"
      },
      "execution_count": 94,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 4. x[n] = n·(u[n] - u[n - 10]), 0 ≤ n ≤ 20"
      ],
      "metadata": {
        "id": "aHy-i2_5fZW-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.arange(0,20+1)\n",
        "\n",
        "x = n*(stepseq(0,0,20)[1] - stepseq(10,0,20)[1])\n",
        "   \n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.title(\"x[n] = n·(u[n] - u[n - 10])\")\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "i6LX3t42fe8n",
        "outputId": "e9e11d2a-29a3-4e9b-89b0-58047479202e"
      },
      "execution_count": 123,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Exponencial real"
      ],
      "metadata": {
        "id": "tZ18rFDMky8F"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 1. x1[n] = (0.8)n, 0 ≤ n ≤ 99"
      ],
      "metadata": {
        "id": "NVzv5WFnmp1W"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.arange(0,100)\n",
        "x = 0.8**n\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "zBjVB_xglxP8",
        "outputId": "7552891e-0eec-4caa-e9a4-4babe362dfa0"
      },
      "execution_count": 130,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 2. x2[n] = (1.5)n, 0 ≤ n ≤ 49"
      ],
      "metadata": {
        "id": "KGfGXAjcmwDE"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.arange(0,50)\n",
        "\n",
        "x = (1.5)**n\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 276
        },
        "id": "_hb2lo3pm7NO",
        "outputId": "7e0e5ac9-dc6c-4513-8f2a-46e1e47961f5"
      },
      "execution_count": 131,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 3. x3[n] = (-0.8)n, 0 ≤ n ≤ 99"
      ],
      "metadata": {
        "id": "_1YB8TDvm-WM"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.arange(0,100)\n",
        "\n",
        "x = (-0.8)**n\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "KwfOKAFSnG0V",
        "outputId": "a47d06f1-3051-46a2-a9f0-cc98ce4a4757"
      },
      "execution_count": 132,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 4. x4[n] = (-1.5)n, 0 ≤ n ≤ 49"
      ],
      "metadata": {
        "id": "MHhzl3vqnI9-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.arange(0,50)\n",
        "\n",
        "x = (-1.5)**n\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 276
        },
        "id": "UPWrxeWEnMQE",
        "outputId": "f36576db-8517-4377-aa46-24c890106ec5"
      },
      "execution_count": 133,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Sinais senoidais"
      ],
      "metadata": {
        "id": "ztUPL1_xozm0"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Gerar os gráficos para a sequência discreta x[n] =2·cos(πn/6), -20 ≤ n ≤ 20"
      ],
      "metadata": {
        "id": "pULAnQU-rBHb"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from math import pi\n",
        "\n",
        "n = np.arange(-20,20+1)\n",
        "\n",
        "x = 2 * np.cos((pi*n)/6)\n",
        "\n",
        "plt.xticks(n[::4])\n",
        "plt.yticks(n[::1])\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.show()\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "1YGQWRwvo4j3",
        "outputId": "461e5cb4-c22e-4bef-cbcc-f7b12a6db7ac"
      },
      "execution_count": 147,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Trace o gráfico das seguintes sequências discretas e determine se estes sinais são periódicos.\n",
        "\n",
        "1. x[n] = cos(3πn/8), 0 ≤ n ≤ 30\n",
        "2. x[n] = cos(n), 0 ≤ n ≤ 30\n",
        "3. x[n] = 5·cos(0.5πn + π/4) + 2·sin(0.5πn), 0 ≤ n ≤ 30"
      ],
      "metadata": {
        "id": "QnM4XGGUrHd6"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        " #### x[n] = cos(3πn/8), 0 ≤ n ≤ 30"
      ],
      "metadata": {
        "id": "LB3DfYMVrTQ3"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.arange(0,31)\n",
        "\n",
        "x = np.cos((3*np.pi*n)/8)\n",
        "\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.title(\"x[n] = cos(3πn/8)\")\n",
        "plt.show()\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "N7IM9vjkrVeD",
        "outputId": "94ba5fb2-04b2-4aa7-cad7-346bf828c56d"
      },
      "execution_count": 156,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### x[n] = cos(n), 0 ≤ n ≤ 30"
      ],
      "metadata": {
        "id": "pTLdgWzUrXfP"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.arange(0,31)\n",
        "\n",
        "x = np.cos(n)\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.title(\"x[n] = cos(n)\")\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "V6HOOHb8raJH",
        "outputId": "aa12a045-d39a-46c3-a699-7c983d344b17"
      },
      "execution_count": 157,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### x[n] = 5·cos(0.5πn + π/4) + 2·sin(0.5πn), 0 ≤ n ≤ 30"
      ],
      "metadata": {
        "id": "FgBV2b9yratI"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.arange(0,31)\n",
        "\n",
        "x = 5*np.cos(0.5*np.pi*n + (np.pi)/4) + 2 * np.sin(0.5 * np.pi * n)\n",
        "\n",
        "plt.stem(n,x,use_line_collection=True)\n",
        "plt.title(\"x[n] = 5·cos(0.5πn + π/4) + 2·sin(0.5πn)\")\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "Ir2vh3BmrePn",
        "outputId": "46b3e910-51ab-491d-e2f4-fb64d898afb2"
      },
      "execution_count": 158,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Convolução"
      ],
      "metadata": {
        "id": "qv-YA-oxuggU"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "x = np.array([0,1,2,3,2,1])\n",
        "g = np.array([1,1,1,1,1,1])\n",
        "\n",
        "conv = np.convolve(x,g)\n",
        "n = np.arange(0,x.size+g.size-1)\n",
        "\n",
        "plt.stem(n,conv,use_line_collection=True)\n",
        "plt.title(\"x1[n] = x[n]*g[n]\")\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "I_ERduQxukX_",
        "outputId": "4c2ad358-3ab0-4dbb-8a8c-d39c4dc3b46d"
      },
      "execution_count": 168,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "H6cGi4WxvFSJ"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}