DegrauImpulsoDiscreto.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "id": "2d4af1f9-d488-4b7e-8f07-7270db48d81e",
      "metadata": {
        "tags": [],
        "id": "2d4af1f9-d488-4b7e-8f07-7270db48d81e"
      },
      "source": [
        "# Sinais degrau e impulso unitário em tempo discreto\n",
        "\n",
        "Renato Naville Watanabe"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "5852e163-e642-4b0f-8f6e-e352d6ca6158",
      "metadata": {
        "id": "5852e163-e642-4b0f-8f6e-e352d6ca6158"
      },
      "source": [
        "## Preparação do ambiente"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "29d45909-6338-4cb5-9398-0723f2e0200f",
      "metadata": {
        "tags": [],
        "id": "29d45909-6338-4cb5-9398-0723f2e0200f"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "23dfb6ee-14a5-49c0-9829-7db3c0529c66",
      "metadata": {
        "tags": [],
        "id": "23dfb6ee-14a5-49c0-9829-7db3c0529c66"
      },
      "source": [
        "## Sinal impulso unitário"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "2e95b19e-00ec-4e34-88bd-b599c03a9e25",
      "metadata": {
        "tags": [],
        "id": "2e95b19e-00ec-4e34-88bd-b599c03a9e25"
      },
      "source": [
        "É o sinal mais simples que existe em tempio discreto. Para representá-lo, usamos a letra $\\delta$, e é definido da seguinte maneira:\n",
        "\n",
        "$$\\delta[n] = \\left\\{\\begin{array}{cc}1& ,n = 0 \\\\ 0& ,c.c \\end{array}\\right. $$\n",
        "\n",
        "O seu gráfico é o seguinte:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "4509d9dc-7741-4398-b2d2-ceb872ed88cd",
      "metadata": {
        "tags": [],
        "id": "4509d9dc-7741-4398-b2d2-ceb872ed88cd",
        "outputId": "dcfdb487-78c8-4877-d078-6bd933f70a00"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "n = np.arange(-10,11)\n",
        "x = np.zeros_like(n)\n",
        "x[n==0] = 1\n",
        "\n",
        "plt.stem(n,x)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('x');"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "d113ffd4-95ad-4c10-be64-7cefe2b5b32e",
      "metadata": {
        "tags": [],
        "id": "d113ffd4-95ad-4c10-be64-7cefe2b5b32e"
      },
      "source": [
        "Para obter um impulso unitário em outra posição que não $n=0$, basta atrasarmos (ou adiantarmos) o sinal $\\delta[n]$. Se queremos que o sinal seja 1 em $n=k$, basta fazermos $\\delta[n-k]$."
      ]
    },
    {
      "cell_type": "markdown",
      "id": "7fcc67b5-cfac-4f14-8c40-2cfbd10c07ee",
      "metadata": {
        "tags": [],
        "id": "7fcc67b5-cfac-4f14-8c40-2cfbd10c07ee"
      },
      "source": [
        "---\n",
        "\n",
        "### Exemplo 1\n",
        "\n",
        "Por exemplo, se queremos um impulso unitário em $n=2$, fazemos $\\delta[n-2]$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "02ba1adc-5603-45a7-96f6-2e3fce814ead",
      "metadata": {
        "tags": [],
        "id": "02ba1adc-5603-45a7-96f6-2e3fce814ead",
        "outputId": "71804e5b-ee6c-4cf6-cc06-d43cea1dc14f"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "n = np.arange(-10,11)\n",
        "x = np.zeros_like(n)\n",
        "x[n==2] = 1\n",
        "\n",
        "plt.stem(n,x)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('x');"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "0c02a26d-e817-4fa6-9d1d-2593d8f53591",
      "metadata": {
        "id": "0c02a26d-e817-4fa6-9d1d-2593d8f53591"
      },
      "source": [
        "---"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "8e925e7d-55de-4408-b486-67d3cc2bd667",
      "metadata": {
        "tags": [],
        "id": "8e925e7d-55de-4408-b486-67d3cc2bd667"
      },
      "source": [
        "## Sinal degrau\n",
        "\n",
        "Outro sinal que usaremos bastante é o sinal degrau de tempo discreto. Esse sinal é sempre igual a 1 para $n\\geq 0$. Para esse sinal usaremos a letra $u$:\n",
        "\n",
        "$$u[n] =   \\left\\{\\begin{array}{cc}1& ,n \\geq 0 \\\\ 0& ,c.c \\end{array}\\right. $$\n",
        "\n",
        "O seu gráfico é mostrado abaixo:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "87c08f03-e0ac-4bfb-9905-b2e70cac8d12",
      "metadata": {
        "tags": [],
        "id": "87c08f03-e0ac-4bfb-9905-b2e70cac8d12",
        "outputId": "f15c6e54-44d8-4f9f-b5e0-29ce27288e1f"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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WIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABglbAIN8uWLVN6erri4uKUnZ2tTZs2dWjdyy+/rKioKI0fPz6wDQIAgIgR8nCzatUqFRUVyel0asuWLcrMzFReXp5qa2vbXbdv3z79x3/8h66//vogdQoAACJByMPN4sWLNXXqVOXn52vYsGEqLS1Vz549tWLFijbXtLS06O6779bcuXN12WWXBbFbAAAQ7kIabpqbm1VVVaXc3FzPWHR0tHJzc1VZWdnmunnz5qlfv36aMmXKOR+jqalJ9fX1XjcAAGCvkIabY8eOqaWlRUlJSV7jSUlJcrlcPte8//77ev7557V8+fIOPUZJSYkSEhI8t7S0tPPuGwAAhK+Qvyzlj4aGBt1zzz1avny5EhMTO7SmuLhYdXV1ntvBgwcD3CUAAAilbqF88MTERMXExKimpsZrvKamRsnJya3mf/LJJ9q3b5/GjRvnGXO73ZKkbt26affu3br88su91jgcDjkcjgB0DwAAwlFIz9zExsZq5MiRKi8v94y53W6Vl5crJyen1fwhQ4boo48+UnV1tef2gx/8QDfeeKOqq6t5yQkAAIT2zI0kFRUVafLkyRo1apSysrK0ZMkSNTY2Kj8/X5I0adIk9e/fXyUlJYqLi1NGRobX+j59+khSq3EAAHBhCnm4mTBhgj777DPNmTNHLpdLI0aMUFlZmeci4wMHDig6OqIuDQIAACEU8nAjSYWFhSosLPS5raKiot21K1eu7PqGAABAxOKUCAAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABgFcINAACwCuEGAABYhXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABgFcINAACwCuEGAABYhXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABglbAIN8uWLVN6erri4uKUnZ2tTZs2tTl3+fLluv7663XRRRfpoosuUm5ubrvzAQDAhSXk4WbVqlUqKiqS0+nUli1blJmZqby8PNXW1vqcX1FRoYkTJ+qdd95RZWWl0tLSdPPNN+vw4cNB7hwAAISjkIebxYsXa+rUqcrPz9ewYcNUWlqqnj17asWKFT7n/+EPf9DPf/5zjRgxQkOGDNHvf/97ud1ulZeXB7lzAAAQjkIabpqbm1VVVaXc3FzPWHR0tHJzc1VZWdmhGidPntTp06fVt29fn9ubmppUX1/vdQMAAPYKabg5duyYWlpalJSU5DWelJQkl8vVoRoPP/ywUlNTvQLS15WUlCghIcFzS0tLO+++AQBA+Ar5y1LnY/78+Xr55Zf1+uuvKy4uzuec4uJi1dXVeW4HDx4McpcAACCYuoXywRMTExUTE6Oamhqv8ZqaGiUnJ7e7dtGiRZo/f77efvttDR8+vM15DodDDoejS/oFAADhL6RnbmJjYzVy5Eivi4HPXhyck5PT5roFCxboscceU1lZmUaNGhWMVgEAQIQI6ZkbSSoqKtLkyZM1atQoZWVlacmSJWpsbFR+fr4kadKkSerfv79KSkokSU8++aTmzJmjP/7xj0pPT/dcm9OrVy/16tUrZPsBAADCQ8jDzYQJE/TZZ59pzpw5crlcGjFihMrKyjwXGR84cEDR0f84wfTss8+qublZP/7xj73qOJ1OPfroo8FsHQAAhKGQhxtJKiwsVGFhoc9tFRUVXvf37dsX+IYAAEDEiuh3SwEAAHwT4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABgFcINAACwCuEGAABYhXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKoQbAABgFcINAACwCuEGAABYhXADAACsQrgBAABWIdwAAACrEG4AAIBVCDcAAMAqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCrdQt2ALVrcRpv2Hldtwyn16x2nrEF9FRMdFbZ1A1k7EnsOZG16Dl5tAJDCJNwsW7ZMCxculMvlUmZmppYuXaqsrKw2569evVqzZ8/Wvn37dMUVV+jJJ5/UrbfeGsSOvZVtO6q5b+3Q0bpTnrGUhDg5xw3T2IyUsKtLz8GrTc/Bqw0AZ4X8ZalVq1apqKhITqdTW7ZsUWZmpvLy8lRbW+tz/oYNGzRx4kRNmTJFW7du1fjx4zV+/Hht27YtyJ1/pWzbUU1/aYvXL2tJctWd0vSXtqhs29GwqkvPwatNz8GrDQBfF/IzN4sXL9bUqVOVn58vSSotLdWaNWu0YsUKzZo1q9X8Z555RmPHjtVDDz0kSXrssce0fv16/eY3v1FpaWlQe29xG819a4eMJBkjR0uz1/YoSSWvbdVN6fF+nXZvcRuVvLZVsWeafG7vbN1A1o7EngNZm56DV/ssd/MZOf5/fffJk3Kf6bpfb4GqTc+RX5ue265tjOmyuv6KMiF89ObmZvXs2VOvvvqqxo8f7xmfPHmyTpw4oTfffLPVmgEDBqioqEgPPPCAZ8zpdOqNN97QX//611bzm5qa1NT0j1+o9fX1SktLU11dneLj48+r/8pPPtfE5R9IkhxnmvTGn395XvUAALBF2gcb1avP+f2d/br6+nolJCR06O93SF+WOnbsmFpaWpSUlOQ1npSUJJfL5XONy+Xya35JSYkSEhI8t7S0tK5pXlJtw6lzTwIA4ALUMzZ0Lw6F/GWpQCsuLlZRUZHn/tkzN12hX+84z383xcRq/O2P+5y3Mj9L2YP6drjuxr3Hde8Lm845z9+6gawdiT0HsjY9B682gPAU1aNHyB47pOEmMTFRMTExqqmp8RqvqalRcnKyzzXJycl+zXc4HHI4HF3T8DdkDeqrlIQ4uepOyURFqamb9+NESUpOiFPW0P6K9uM6gqyhPdT34oSv6vrY3tm6gawdiT0HsjY9B682AHxTSF+Wio2N1ciRI1VeXu4Zc7vdKi8vV05Ojs81OTk5XvMlaf369W3OD6SY6Cg5xw2T9NUv5687e985bpjfF0gGqm4ga0diz4GsTc/Bqw0A3xTyt4IXFRVp+fLlevHFF7Vz505Nnz5djY2NnndPTZo0ScXFxZ75M2bMUFlZmZ566int2rVLjz76qDZv3qzCwsKQ9D82I0XP/vQaJSfEeY0nJ8Tp2Z9e0+nP7ghUXXoOXm16Dl5tAPi6kL5b6qzf/OY3ng/xGzFihH79618rOztbkjRmzBilp6dr5cqVnvmrV6/WI4884vkQvwULFnT4Q/z8udraH3xSbODrRmpteg5ebQD28ufvd1iEm2AKVLgBAACBEzFvBQcAAOhqhBsAAGAVwg0AALAK4QYAAFiFcAMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCoh/VbwUDj7gcz19fUh7gQAAHTU2b/bHflihQsu3DQ0NEiS0tLSQtwJAADwV0NDgxISEtqdc8F9t5Tb7daRI0fUu3dvRUV17Zf11dfXKy0tTQcPHrTye6ts3z/J/n1k/yKf7fvI/kW+QO2jMUYNDQ1KTU1VdHT7V9VccGduoqOjdemllwb0MeLj4639n1ayf/8k+/eR/Yt8tu8j+xf5ArGP5zpjcxYXFAMAAKsQbgAAgFUIN13I4XDI6XTK4XCEupWAsH3/JPv3kf2LfLbvI/sX+cJhHy+4C4oBAIDdOHMDAACsQrgBAABWIdwAAACrEG4AAIBVCDd+ePzxxzV69Gj17NlTffr08TnnwIEDuu2229SzZ0/169dPDz30kM6cOdNu3ePHj+vuu+9WfHy8+vTpoylTpuiLL74IwB74p6KiQlFRUT5vH374YZvrxowZ02r+tGnTgth5x6Wnp7fqdf78+e2uOXXqlAoKCnTxxRerV69e+ud//mfV1NQEqWP/7Nu3T1OmTNGgQYPUo0cPXX755XI6nWpubm53XTgfw2XLlik9PV1xcXHKzs7Wpk2b2p2/evVqDRkyRHFxcbrqqqu0du3aIHXqv5KSEl177bXq3bu3+vXrp/Hjx2v37t3trlm5cmWrYxUXFxekjv3z6KOPtup1yJAh7a6JpOMn+f6dEhUVpYKCAp/zw/34vffeexo3bpxSU1MVFRWlN954w2u7MUZz5sxRSkqKevToodzcXP39738/Z11/n8f+Itz4obm5WXfeeaemT5/uc3tLS4tuu+02NTc3a8OGDXrxxRe1cuVKzZkzp926d999t7Zv367169frz3/+s9577z3df//9gdgFv4wePVpHjx71ut13330aNGiQRo0a1e7aqVOneq1bsGBBkLr237x587x6/bd/+7d258+cOVNvvfWWVq9erXfffVdHjhzRj370oyB1659du3bJ7Xbrueee0/bt2/X000+rtLRUv/jFL865NhyP4apVq1RUVCSn06ktW7YoMzNTeXl5qq2t9Tl/w4YNmjhxoqZMmaKtW7dq/PjxGj9+vLZt2xbkzjvm3XffVUFBgT744AOtX79ep0+f1s0336zGxsZ218XHx3sdq/379wepY/995zvf8er1/fffb3NupB0/Sfrwww+99m/9+vWSpDvvvLPNNeF8/BobG5WZmally5b53L5gwQL9+te/VmlpqTZu3KhvfetbysvL06lTp9qs6e/zuFMM/PbCCy+YhISEVuNr16410dHRxuVyecaeffZZEx8fb5qamnzW2rFjh5FkPvzwQ8/Yf//3f5uoqChz+PDhLu/9fDQ3N5tLLrnEzJs3r915N9xwg5kxY0ZwmjpPAwcONE8//XSH5584ccJ0797drF692jO2c+dOI8lUVlYGoMOut2DBAjNo0KB254TrMczKyjIFBQWe+y0tLSY1NdWUlJT4nP8v//Iv5rbbbvMay87ONj/72c8C2mdXqa2tNZLMu+++2+actn4fhSOn02kyMzM7PD/Sj58xxsyYMcNcfvnlxu12+9weScdPknn99dc9991ut0lOTjYLFy70jJ04ccI4HA7zX//1X23W8fd53BmcuelClZWVuuqqq5SUlOQZy8vLU319vbZv397mmj59+nidCcnNzVV0dLQ2btwY8J798ac//Umff/658vPzzzn3D3/4gxITE5WRkaHi4mKdPHkyCB12zvz583XxxRfr6quv1sKFC9t9GbGqqkqnT59Wbm6uZ2zIkCEaMGCAKisrg9Hueaurq1Pfvn3POS/cjmFzc7Oqqqq8fvbR0dHKzc1t82dfWVnpNV/66jkZScdK0jmP1xdffKGBAwcqLS1NP/zhD9v8fRMO/v73vys1NVWXXXaZ7r77bh04cKDNuZF+/Jqbm/XSSy/pX//1X9v9ouZIOn5ft3fvXrlcLq9jlJCQoOzs7DaPUWeex51xwX1xZiC5XC6vYCPJc9/lcrW5pl+/fl5j3bp1U9++fdtcEyrPP/+88vLyzvnFo3fddZcGDhyo1NRU/e1vf9PDDz+s3bt367XXXgtSpx337//+77rmmmvUt29fbdiwQcXFxTp69KgWL17sc77L5VJsbGyra66SkpLC7nj5smfPHi1dulSLFi1qd144HsNjx46ppaXF53Ns165dPte09ZyMhGPldrv1wAMP6LrrrlNGRkab8wYPHqwVK1Zo+PDhqqur06JFizR69Ght37494F8S7K/s7GytXLlSgwcP1tGjRzV37lxdf/312rZtm3r37t1qfiQfP0l64403dOLECd17771tzomk4/dNZ4+DP8eoM8/jzrjgw82sWbP05JNPtjtn586d57zoLZJ0Zp8PHTqkdevW6ZVXXjln/a9fL3TVVVcpJSVFN910kz755BNdfvnlnW+8g/zZv6KiIs/Y8OHDFRsbq5/97GcqKSkJ649H78wxPHz4sMaOHas777xTU6dObXdtqI8hpIKCAm3btq3da1IkKScnRzk5OZ77o0eP1tChQ/Xcc8/pscceC3Sbfrnllls8/z18+HBlZ2dr4MCBeuWVVzRlypQQdhYYzz//vG655Ralpqa2OSeSjl8kueDDzYMPPthuqpakyy67rEO1kpOTW13xffZdNMnJyW2u+eZFVGfOnNHx48fbXHO+OrPPL7zwgi6++GL94Ac/8PvxsrOzJX111iAYfxjP55hmZ2frzJkz2rdvnwYPHtxqe3Jyspqbm3XixAmvszc1NTUBO16++LuPR44c0Y033qjRo0frd7/7nd+PF+xj6EtiYqJiYmJavTOtvZ99cnKyX/PDRWFhoefNBf7+67179+66+uqrtWfPngB113X69OmjK6+8ss1eI/X4SdL+/fv19ttv+322M5KO39njUFNTo5SUFM94TU2NRowY4XNNZ57HndJlV+9cQM51QXFNTY1n7LnnnjPx8fHm1KlTPmudvaB48+bNnrF169aF1QXFbrfbDBo0yDz44IOdWv/+++8bSeavf/1rF3fW9V566SUTHR1tjh8/7nP72QuKX331Vc/Yrl27wvqC4kOHDpkrrrjC/OQnPzFnzpzpVI1wOYZZWVmmsLDQc7+lpcX079+/3QuKb7/9dq+xnJycsL0g1e12m4KCApOammo+/vjjTtU4c+aMGTx4sJk5c2YXd9f1GhoazEUXXWSeeeYZn9sj7fh9ndPpNMnJyeb06dN+rQvn46c2LihetGiRZ6yurq5DFxT78zzuVK9dVukCsH//frN161Yzd+5c06tXL7N161azdetW09DQYIz56n/KjIwMc/PNN5vq6mpTVlZmLrnkElNcXOypsXHjRjN48GBz6NAhz9jYsWPN1VdfbTZu3Gjef/99c8UVV5iJEycGff/a8vbbbxtJZufOna22HTp0yAwePNhs3LjRGGPMnj17zLx588zmzZvN3r17zZtvvmkuu+wy893vfjfYbZ/Thg0bzNNPP22qq6vNJ598Yl566SVzySWXmEmTJnnmfHP/jDFm2rRpZsCAAeZ///d/zebNm01OTo7JyckJxS6c06FDh8y3v/1tc9NNN5lDhw6Zo0ePem5fnxMpx/Dll182DofDrFy50uzYscPcf//9pk+fPp53KN5zzz1m1qxZnvl/+ctfTLdu3cyiRYvMzp07jdPpNN27dzcfffRRqHahXdOnTzcJCQmmoqLC61idPHnSM+eb+zh37lyzbt0688knn5iqqirzk5/8xMTFxZnt27eHYhfa9eCDD5qKigqzd+9e85e//MXk5uaaxMREU1tba4yJ/ON3VktLixkwYIB5+OGHW22LtOPX0NDg+VsnySxevNhs3brV7N+/3xhjzPz5802fPn3Mm2++af72t7+ZH/7wh2bQoEHmyy+/9NT43ve+Z5YuXeq5f67ncVcg3Phh8uTJRlKr2zvvvOOZs2/fPnPLLbeYHj16mMTERPPggw96Jfd33nnHSDJ79+71jH3++edm4sSJplevXiY+Pt7k5+d7AlM4mDhxohk9erTPbXv37vX6GRw4cMB897vfNX379jUOh8N8+9vfNg899JCpq6sLYscdU1VVZbKzs01CQoKJi4szQ4cONU888YTXWbZv7p8xxnz55Zfm5z//ubnoootMz549zR133OEVFsLJCy+84PP/2a+ftI20Y7h06VIzYMAAExsba7KysswHH3zg2XbDDTeYyZMne81/5ZVXzJVXXmliY2PNd77zHbNmzZogd9xxbR2rF154wTPnm/v4wAMPeH4eSUlJ5tZbbzVbtmwJfvMdMGHCBJOSkmJiY2NN//79zYQJE8yePXs82yP9+J21bt06I8ns3r271bZIO35n/2Z983Z2H9xut5k9e7ZJSkoyDofD3HTTTa32e+DAgcbpdHqNtfc87gpRxhjTdS9yAQAAhBafcwMAAKxCuAEAAFYh3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAq3QLdQMAcL7GjBmj4cOHKy4uTr///e8VGxuradOm6dFHHw11awBCgDM3AKzw4osv6lvf+pY2btyoBQsWaN68eVq/fn2o2wIQAnwrOICIN2bMGLW0tOj//u//PGNZWVn63ve+p/nz54ewMwChwJkbAFYYPny41/2UlBTV1taGqBsAoUS4AWCF7t27e92PioqS2+0OUTcAQolwAwAArEK4AQAAViHcAAAAq/BuKQAAYBXO3AAAAKsQbgAAgFUINwAAwCqEGwAAYBXCDQAAsArhBgAAWIVwAwAArEK4AQAAViHcAAAAqxBuAACAVQg3AADAKv8Pfga7maM6hKIAAAAASUVORK5CYII=",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "n = np.arange(-10,11)\n",
        "x = np.zeros_like(n)\n",
        "x[n>=0] = 1\n",
        "\n",
        "plt.stem(n,x)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('x');"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "4a0b34a2-4d71-4816-acff-7023fd995027",
      "metadata": {
        "tags": [],
        "id": "4a0b34a2-4d71-4816-acff-7023fd995027"
      },
      "source": [
        "Os sinais degrau e impulso unitário em tempo discreto estão bastante relacionados. O sinal impulso unitário é o sinal degrau subtraido do mesmo sinal degrau atrasado de 1 passo.\n",
        "\n",
        "$$\\delta[n] = u[n] - u[n-1]$$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "2f4d6d7d-7bbe-49ca-a0d7-3b9becffd22a",
      "metadata": {
        "tags": [],
        "id": "2f4d6d7d-7bbe-49ca-a0d7-3b9becffd22a"
      },
      "source": [
        "## Exercícios\n",
        "\n",
        "**1)** Faça o gráfico dos seguintes  sinais:\n",
        "\n",
        "a) $x_1[n]=u[n] - u[n-3]$\n",
        "\n",
        "b) $x_2[n] = u[n] - 2u[n-3]+u[n-4]$\n",
        "\n",
        "c) $x_3[n] = nu[n]-2(n-1)u[n-1]+(n-3)u[n-3]$\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "f965ad92-2c15-46fe-a9af-85e3dee97b22",
      "metadata": {
        "tags": [],
        "id": "f965ad92-2c15-46fe-a9af-85e3dee97b22"
      },
      "source": [
        "**2)** Faça o gráfico dos seguintes sinais:\n",
        "\n",
        "a) $x_1[n] = u[n] - \\delta[n-3]-u[n-4]$\n",
        "\n",
        "b) $x_2[n] = nu[n] - nu[n-5] + 3\\delta[n-2]$\n",
        "\n",
        "c) $x_3[n] = 2\\delta[n-1] - 4\\delta[n-3]$"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "0129c8a2-de0e-478b-b267-9c8c0dd8dcb6",
      "metadata": {
        "tags": [],
        "id": "0129c8a2-de0e-478b-b267-9c8c0dd8dcb6"
      },
      "source": [
        "**3)** Considere o seguinte sinal de tempo discreto:\n",
        "\n",
        "$$x[n] = 1-\\displaystyle\\sum_{k=2}^{\\infty}\\delta[n-2-k] $$\n",
        "\n",
        "Determine os valores dos números inteiros $M$ e $n_0$ para escrever $x[n]$ na forma:\n",
        "\n",
        "$$x[n]=u[Mn-n_0] $$"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "4ad17ae3-b797-4a0a-afe4-c641ab91106f",
      "metadata": {
        "tags": [],
        "id": "4ad17ae3-b797-4a0a-afe4-c641ab91106f"
      },
      "source": [
        "**4)** Considere o seguinte sinal:\n",
        "\n",
        "$$x[n] = 0.5^{n-1}(u[n+2]- u[n-8])$$\n",
        "\n",
        "Encontre $a$ e $b$ na expressão abaixo:\n",
        "\n",
        "$$x[n-k] = \\left\\{\\begin{array}{cc}0.5^{n-k-1}&\\text{ , se } a< k < b \\\\ 0& \\text{ , c.c.} \\end{array}\\right. $$"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "67555f96-2adf-4788-bd8b-9e7dbfb6b9f9",
      "metadata": {
        "id": "67555f96-2adf-4788-bd8b-9e7dbfb6b9f9"
      },
      "source": [
        "**5)** Escreva os seguintes sinais com apenas uma expressão (sem o uso de chaves com condições):\n",
        "\n",
        "a) $x_1[n] = \\left\\{\\begin{array}{cc} n & , 0 \\leq n \\leq 3\\\\n^2&, 3< n \\leq 7\\\\ 0& \\text{c.c.} \\end{array}\\right.$\n",
        "\n",
        "\n",
        "b) $x_2[n] = \\left\\{\\begin{array}{cc} \\cos\\left(\\frac{\\pi}{6}n\\right) & , -3 \\leq n \\leq 3\\\\ 0&\\text{c.c.} \\end{array}\\right.$\n",
        "\n",
        "$x[n] = \\left\\{\\begin{array}{cc} 1 & , 0 \\leq n \\leq 2\\\\2&, 2< n \\leq 7\\\\ 1&7 < n \\leq 10 \\\\0&\\text{c.c.} \\end{array}\\right.$"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "3756de4b-629f-47a9-be93-050e708a3b64",
      "metadata": {
        "tags": [],
        "id": "3756de4b-629f-47a9-be93-050e708a3b64"
      },
      "source": [
        "## Referências\n",
        "\n",
        "- Oppenheim, M.  e Willsky, S., Sistemas e Sinais, 2010\n",
        "- Ulaby, FT e Yagle, AE, Signals & Systems: Theory and Applications, 2018, [ss2.eecs.umich.edu](ss2.eecs.umich.edu)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "8968544a-1148-4dc0-9c85-2b462776831f",
      "metadata": {
        "id": "8968544a-1148-4dc0-9c85-2b462776831f"
      },
      "source": [
        "## Solução dos exercícios"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "23b4f3cf-179b-4d06-936b-312ce88c7cf2",
      "metadata": {
        "tags": [],
        "id": "23b4f3cf-179b-4d06-936b-312ce88c7cf2"
      },
      "source": [
        "**1-)** Resolvido por Frederico Guazzelli"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "37572215-b377-4a55-bf09-f2a98229176d",
      "metadata": {
        "tags": [],
        "id": "37572215-b377-4a55-bf09-f2a98229176d",
        "outputId": "5653a943-7a05-4dc2-f8ba-6eb931c7cc7a"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "/tmp/ipykernel_56971/1118574286.py:8: MatplotlibDeprecationWarning: The 'use_line_collection' parameter of stem() was deprecated in Matplotlib 3.6 and will be removed two minor releases later. If any parameter follows 'use_line_collection', they should be passed as keyword, not positionally.\n",
            "  plt.stem(n, x1, use_line_collection=True)\n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#1a)\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "n = np.arange(0, 10)\n",
        "x1 = np.heaviside(n, 1) - np.heaviside(n - 3, 1)\n",
        "\n",
        "plt.stem(n, x1, use_line_collection=True)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('Amplitude')\n",
        "plt.title(r'$x_1[n] = u[n] - u[n-3]$')\n",
        "\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "1223983f-8042-4a5b-abcf-1cfeea56b5fa",
      "metadata": {
        "tags": [],
        "id": "1223983f-8042-4a5b-abcf-1cfeea56b5fa",
        "outputId": "70adce3d-2ec1-41d2-eb0e-ce0d0535508d"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "/tmp/ipykernel_52260/2640829037.py:6: MatplotlibDeprecationWarning: The 'use_line_collection' parameter of stem() was deprecated in Matplotlib 3.6 and will be removed two minor releases later. If any parameter follows 'use_line_collection', they should be passed as keyword, not positionally.\n",
            "  plt.stem(n, x2, use_line_collection=True)\n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#1b)\n",
        "n = np.arange(0, 10)\n",
        "\n",
        "x2 = np.heaviside(n, 1) - 2 * np.heaviside(n - 3, 1) + np.heaviside(n - 4, 1)\n",
        "\n",
        "plt.stem(n, x2, use_line_collection=True)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('Amplitude')\n",
        "plt.title(r'$x_2[n] = u[n] - 2u[n-3] + u[n-4]$')\n",
        "\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "90bf54b1-110b-4024-a86e-ad5fcce109ef",
      "metadata": {
        "tags": [],
        "id": "90bf54b1-110b-4024-a86e-ad5fcce109ef",
        "outputId": "650b8a6d-4eec-4b5b-ac21-6be0b6d326ee"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "/tmp/ipykernel_52260/3793139166.py:6: MatplotlibDeprecationWarning: The 'use_line_collection' parameter of stem() was deprecated in Matplotlib 3.6 and will be removed two minor releases later. If any parameter follows 'use_line_collection', they should be passed as keyword, not positionally.\n",
            "  plt.stem(n, x3, use_line_collection=True)\n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#1c)\n",
        "n = np.arange(0, 10)\n",
        "\n",
        "x3 = n * np.heaviside(n, 1) - 2 * (n - 1) * np.heaviside(n - 1, 1) + (n - 3) * np.heaviside(n - 3, 1)\n",
        "\n",
        "plt.stem(n, x3, use_line_collection=True)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('Amplitude')\n",
        "plt.title(r'$x_3[n] = nu[n] - 2(n-1)u[n-1] + (n-3)u[n-3]$')\n",
        "\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "90a995e1-b677-4bb4-9d2d-75ca21fa2b70",
      "metadata": {
        "id": "90a995e1-b677-4bb4-9d2d-75ca21fa2b70"
      },
      "source": [
        "**2-)** Resolvido por Frederico Guazzelli"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "00c5057e-a91c-4aa3-89a5-a99e4266f449",
      "metadata": {
        "tags": [],
        "id": "00c5057e-a91c-4aa3-89a5-a99e4266f449",
        "outputId": "c49cad1f-b871-475a-b496-467d0c4b2f50"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "/tmp/ipykernel_56971/2201575927.py:5: MatplotlibDeprecationWarning: The 'use_line_collection' parameter of stem() was deprecated in Matplotlib 3.6 and will be removed two minor releases later. If any parameter follows 'use_line_collection', they should be passed as keyword, not positionally.\n",
            "  plt.stem(n, x1, use_line_collection=True)\n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#2a)\n",
        "n = np.arange(0, 10)\n",
        "x1 = np.heaviside(n, 1) - (n == 3) - np.heaviside(n - 4, 1)\n",
        "\n",
        "plt.stem(n, x1, use_line_collection=True)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('Amplitude')\n",
        "plt.title(r'$x_1[n] = u[n] - \\delta[n-3] - u[n-4]$')\n",
        "\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "b7de82b3-7925-4acc-83a6-995b0f4562b6",
      "metadata": {
        "tags": [],
        "id": "b7de82b3-7925-4acc-83a6-995b0f4562b6",
        "outputId": "fa102aaa-b628-4382-f9a8-72875518ad97"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "/tmp/ipykernel_56971/3654669483.py:6: MatplotlibDeprecationWarning: The 'use_line_collection' parameter of stem() was deprecated in Matplotlib 3.6 and will be removed two minor releases later. If any parameter follows 'use_line_collection', they should be passed as keyword, not positionally.\n",
            "  plt.stem(n, x2, use_line_collection=True)\n"
          ]
        },
        {
          "data": {
            "image/png": 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73S673V7DVcEKia0aK9YRrvyi0grPU7FJinGcu1QZAFB3WX6OyoVOnDihvXv3KjaWy06DXWiITalJ8ZLOhZILnV9OTYrnRFoAqOMsDSp/+MMftHr1au3bt0/r16/XPffco9DQUA0bNszKsmCIgQmxSh/RRdENvWfRYhzhSh/RhfuoAEAQsPTQz48//qhhw4bp2LFjioqK0s0336yNGzcqKirKyrJgkIEJserduok6PPW5JGnRqO7qc0MUMykAECQsDSpLliyxcvOoJS4MJdw+HwCCi1HnqAAAAFyIoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYCyCCgAAMBZBBQAAGIugAgAAjEVQAQAAxiKoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADAWQQUAABiLoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYCyCCgAAMBZBBQAAGIugAgAAjEVQAQAAxiKoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADAWQQUAABiLoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYKwrrC4AQO3idLm1OadQBSWlio4MV2KrxgoNsVldFoA6ypgZlRkzZshms2ny5MlWlwLgIpZn5enmmV9q2IKNmrQkU8MWbNTNM7/U8qw8q0sDUEcZEVS2bNmi1157TR07drS6FAAXsTwrT+MWZyivqNSrPb+oVOMWZxBWAFQLy4PKiRMnNHz4cC1YsECNGjWyuhwAFXC63Jq2LFvuCp473zZtWbacrop6AEDlWR5Uxo8fr7vuuksDBgy4bN+ysjIVFxd7PQBUv805heVmUi7klpRXVKrNOYU1VxSAoGDpybRLlixRRkaGtmzZ4lP/tLQ0TZs2rZqrAvBLBSUXDymV6QcAvrJsRiU3N1eTJk3SW2+9pfDwcJ9ek5KSoqKiIs8jNze3mqsEIEnRkb59Rn3tBwC+smxGZevWrSooKFCXLl08bU6nU2vWrNHcuXNVVlam0NBQr9fY7XbZ7faaLhUIeomtGivWEa78otIKz1OxSYpxnLtUGQACybIZlf79+2vHjh3KzMz0PLp166bhw4crMzOzXEgBYJ3QEJtSk+IlnQslFzq/nJoUz/1UAAScZTMqkZGRSkhI8Gpr0KCBrr766nLtAKw3MCFW6SO6KHXpdzpcXOZpj3GEKzUpXgMTYi2sDkBdxZ1pAfhsYEKserduog5PfS5JWjSqu/rcEMVMCoBqY1RQWbVqldUlALiMC0MJt88HUN0sv48KAADAxRBUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADAWQQUAABiLoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYCyCCgAAMBZBBQAAGIugAgAAjEVQAQAAxiKoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADAWQQUAABiLoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYCyCCgAAMBZBBQAAGIugAgAAjEVQAQAAxiKoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMdYXVBQDBwulya3NOoQpKShUdGa7EVo0VGmKzuiwAMJqlQSU9PV3p6enat2+fJKl9+/Z68sknNWjQICvLAgJueVaepi3LVl5Rqact1hGu1KR4DUyItbAyADCbpYd+WrRooRkzZmjr1q365ptvdNttt+k3v/mNvvvuOyvLAgJqeVaexi3O8AopkpRfVKpxizO0PCvPosoAwHyWBpWkpCTdeeeduuGGG9SmTRtNnz5dV155pTZu3GhlWUDAOF1uTVuWLXcFz51vm7YsW05XRT0AAMaco+J0OvX+++/r5MmT6tmzZ4V9ysrKVFZW5lkuLi6uqfKAStmcU1huJuVCbkl5RaXanFOontdfXXOFAUAtYflVPzt27NCVV14pu92u3//+9/rwww8VHx9fYd+0tDQ5HA7PIy4uroarBfxTUHLxkFKZfgAQbCoVVI4fP66FCxcqJSVFhYWFkqSMjAwdPHjQ73W1bdtWmZmZ2rRpk8aNG6eRI0cqOzu7wr4pKSkqKiryPHJzcytTPlBjoiPDA9oPAIKN34d+tm/frgEDBsjhcGjfvn0aM2aMGjdurA8++EAHDhzQ3/72N7/WFxYWptatW0uSunbtqi1btugvf/mLXnvttXJ97Xa77Ha7vyUDlkls1VixjnDlF5VWeJ6KTVKM49ylygCA8vyeUUlOTtbDDz+sPXv2KDz8//0v8M4779SaNWuqXJDL5fI6DwWozUJDbEpNOnco85d3TDm/nJoUz/1UAOAi/A4qW7Zs0dixY8u1N2/eXPn5+X6tKyUlRWvWrNG+ffu0Y8cOpaSkaNWqVRo+fLi/ZQHGGpgQq/QRXRTd0Hs2MMYRrvQRXbiPCgBcgt+Hfux2e4VX2+zevVtRUVF+raugoEAPPfSQ8vLy5HA41LFjR3322We6/fbb/S0LMNrAhFj1bt1EHZ76XJK0aFR39bkhipkUALgMv4PK4MGD9fTTT+u9996TJNlsNh04cEB/+tOfdN999/m1rtdff93fzQO11oWhhNvnA4Bv/D70M2vWLJ04cULR0dE6deqUbrnlFrVu3VqRkZGaPn16ddQIAACClN8zKg6HQytWrNC6deu0fft2nThxQl26dNGAAQOqoz4AABDEKn1n2ptvvlk333xzIGsBAADw4lNQefnll31e4cSJEytdDAAAwIV8Ciovvvii1/KRI0f0888/66qrrpJ07k619evXV3R0NEEFAAAEjE8n0+bk5Hge06dP14033qidO3eqsLBQhYWF2rlzp7p06aJnnnmmuusFAABBxO+rfv785z9rzpw5atu2raetbdu2evHFF/W///u/AS0OAAAEN7+DSl5ens6ePVuu3el06vDhwwEpCgAAQKpEUOnfv7/Gjh2rjIwMT9vWrVs1btw4LlEGAAAB5XdQeeONNxQTE6Nu3bp5fs04MTFRTZs21cKFC6ujRgAAEKT8vo9KVFSU/v3vf2v37t3atWuXJKldu3Zq06ZNwIsDAADBrdI3fGvTpg3hBAAAVCu/g8ro0aMv+fwbb7xR6WIAAAAu5HdQ+emnn7yWz5w5o6ysLB0/fly33XZbwAoDAADwO6h8+OGH5dpcLpfGjRun66+/PiBFAQAASJW46qfClYSEKDk5udyt9gEAAKoiIEFFkvbu3VvhjeAAAAAqy+9DP8nJyV7LbrdbeXl5+te//qWRI0cGrDAAAAC/g8q3337rtRwSEqKoqCjNmjXrslcEAQAA+MPvoPLVV19VRx0AAADl+H2Oym233abjx4+Xay8uLubyZAAAEFB+B5VVq1bp9OnT5dpLS0u1du3agBQFAAAg+XHoZ/v27Z4/Z2dnKz8/37PsdDq1fPlyNW/ePLDVAQCAoOZzULnxxhtls9lks9kqPMQTERGhOXPmBLQ4AAAQ3HwOKjk5OXK73bruuuu0efNmRUVFeZ4LCwtTdHS0QkNDq6VIAAAQnHwOKtdee62kc7fLBwAAqAk+BZWlS5dq0KBBqlevnpYuXXrJvoMHDw5IYQAAAD4FlSFDhig/P1/R0dEaMmTIRfvZbDY5nc5A1QYAAIKcT0HlwsM9HPoBAAA1JWA/SggAABBoPs2ovPzyyz6vcOLEiZUuBgAA4EI+BZUXX3zRp5XZbDaCCgAACBifgkpOTk511wEAAFBOlc5RcbvdcrvdgaoFAADAS6WCyuuvv66EhASFh4crPDxcCQkJWrhwYaBrAwAAQc7nO9Oe9+STT2r27Nl65JFH1LNnT0nShg0b9Oijj+rAgQN6+umnA14kAAAITn4HlfT0dC1YsEDDhg3ztA0ePFgdO3bUI488QlABAAAB4/ehnzNnzqhbt27l2rt27aqzZ88GpCgAAACpEkHlv/7rv5Senl6uff78+Ro+fHhAigIAAJAqcehHOncy7eeff66bbrpJkrRp0yYdOHBADz30kJKTkz39Zs+eHZgqAQBAUPI7qGRlZalLly6SpL1790qSmjRpoiZNmigrK8vTz2azBahEAAAQrPwOKl999VV11AEAAFAOP0oIAACM5feMSmlpqebMmaOvvvpKBQUFcrlcXs9nZGQErDgAABDc/A4qv/vd7/T555/r/vvvV2JiIueiAACAauN3UPnkk0/073//W717966OegAAADz8PkelefPmioyMrI5aAAAAvPgdVGbNmqU//elP2r9/f3XUAwAA4OH3oZ9u3bqptLRU1113nerXr6969ep5PV9YWBiw4gAAQHDzO6gMGzZMBw8e1LPPPqumTZtyMi0AAKg2fgeV9evXa8OGDerUqVN11AMAAODh9zkq7dq106lTp6qjFgAAAC9+B5UZM2ZoypQpWrVqlY4dO6bi4mKvBwAAQKD4fehn4MCBkqT+/ft7tbvdbtlsNjmdzsBUBgAAgl5Af5Rwx44dVSoGAADgQn4HlVtuucVruaSkRO+8844WLlyorVu3asKECQErDgAABLdK/3rymjVrNHLkSMXGxuqFF17Qbbfdpo0bN/q1jrS0NHXv3l2RkZGKjo7WkCFD9P3331e2JAAAUMf4NaOSn5+vRYsW6fXXX1dxcbEeeOABlZWV6aOPPlJ8fLzfG1+9erXGjx+v7t276+zZs3r88cd1xx13KDs7Ww0aNPB7fQAAoG7xOagkJSVpzZo1uuuuu/TSSy9p4MCBCg0N1bx58yq98eXLl3stL1q0SNHR0dq6dav69u1b6fUCAIC6weeg8umnn2rixIkaN26cbrjhhmoppqioSJLUuHHjalk/AACoXXw+R2XdunUqKSlR165d1aNHD82dO1dHjx4NWCEul0uTJ09W7969lZCQUGGfsrIy7tsCAEAQ8Tmo3HTTTVqwYIHy8vI0duxYLVmyRM2aNZPL5dKKFStUUlJSpULGjx+vrKwsLVmy5KJ90tLS5HA4PI+4uLgqbRMAAJjN76t+GjRooNGjR2vdunXasWOHpkyZohkzZig6OlqDBw+uVBETJkzQJ598oq+++kotWrS4aL+UlBQVFRV5Hrm5uZXaHgAAqB0qfXmyJLVt21bPPfecfvzxR73zzjt+v97tdmvChAn68MMP9eWXX6pVq1aX7G+329WwYUOvBwAAqLv8vuFbRUJDQzVkyBANGTLEr9eNHz9eb7/9tj7++GNFRkYqPz9fkuRwOBQRERGI0gAAQC1WpRmVqkpPT1dRUZH69eun2NhYz+Pdd9+1siwAAGCIgMyoVJbb7bZy8wAAwHCWzqgAAABcCkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADAWQQUAABiLoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYCyCCgAAMBZBBQAAGIugAgAAjEVQAQAAxiKoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADAWQQUAABiLoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYCyCCgAAMBZBBQAAGIugAgAAjEVQAQAAxiKoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsS4PKmjVrlJSUpGbNmslms+mjjz6yspw6x+lya8PeY/o486A27D0mp8ttdUkAAPjlCis3fvLkSXXq1EmjR4/Wvffea2Updc7yrDxNW5atvKJST1usI1ypSfEamBBrYWUAAPjO0qAyaNAgDRo0yMoS6qTlWXkatzhDv5w/yS8q1bjFGUof0YWwAgCoFThHpY5xutyatiy7XEiR5Gmbtiybw0AAgFqhVgWVsrIyFRcXez3gbXNOodfhnl9yS8orKtXmnMKaKwoAgEqqVUElLS1NDofD84iLi7O6JOMUlFw8pFSmHwAAVqpVQSUlJUVFRUWeR25urtUlGSc6Mjyg/QAAsJKlJ9P6y263y263W12G0RJbNVasI1z5RaUVnqdikxTjCFdiq8Y1XRoAAH6zdEblxIkTyszMVGZmpiQpJydHmZmZOnDggJVl1WqhITalJsVLOhdKLnR+OTUpXqEhv3wWAADzWBpUvvnmG3Xu3FmdO3eWJCUnJ6tz58568sknrSyr1huYEKv0EV0U3dB79inGEc6lyQCAWsXSQz/9+vWT281lstVhYEKserduog5PfS5JWjSqu/rcEMVMCgCgVqlVJ9PCPxeGksRWjQkpAIBah6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADAWQQUAABiLoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYCyCCgAAMBZBBQAAGIugAgAAjEVQAQAAxiKoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADAWQQUAABiLoAIAAIxFUAEAAMYiqAAAAGMRVAAAgLEIKgAAwFgEFQAAYCyCCgAAMBZBBQAAGIugAgAAjEVQAQAAxiKoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADDWFVYXYCKny63NOYUqKClVdGS4Els1VmiIzeqyAARQXfmcsx/mqSv7Ysp+GBFUXnnlFT3//PPKz89Xp06dNGfOHCUmJlpSy/KsPE1blq28olJPW6wjXKlJ8RqYEGtJTQACq658ztkP89SVfTFpPyw/9PPuu+8qOTlZqampysjIUKdOnfTrX/9aBQUFNV7L8qw8jVuc4fXGSFJ+UanGLc7Q8qy8Gq8JQGDVlc85+2GeurIvpu2H5TMqs2fP1pgxYzRq1ChJ0rx58/Svf/1Lb7zxhqZOnVpjdThdbk1bli23JLndsjtPez1vk5T2wbfq37JhrZnCc50+K/vZsnN//vlnuc5a/nZXCvthltq8H06XW2kffKuw/7/+X6otn3P2wzx1ZV8uth9loWFy22yySZq2LFu3x8fU2H7Y3G63u0a2VIHTp0+rfv36+sc//qEhQ4Z42keOHKnjx4/r448/9upfVlamsrL/N3jFxcWKi4tTUVGRGjZsWKVaNuw9pmELNkqS7GfL9NEnT1RpfQAA1BVD7p6usivsnuV3xtykntdfXen1FRcXy+Fw+PT9bemhn6NHj8rpdKpp06Ze7U2bNlV+fn65/mlpaXI4HJ5HXFxcwGopKCm9fCcAAFCj35m1Z85WUkpKipKTkz3L52dUAiE6Mtzz57LQMA25e3qF/RaNSlSPVo0Dsk0ANWtTTqEefnPzZfuZ/jlnP8xTV/blYvtRFhrmtXzhd2Z1szSoNGnSRKGhoTp8+LBX++HDhxUTE1Ouv91ul91uL9ceCImtGivWEa78olK5bTavKS7p3PHFGEe4En/VXCEGH18EcHGJv4pQ46sd5z7nFTxfWz7n7Id56sq++LwfNRi2LD30ExYWpq5du2rlypWeNpfLpZUrV6pnz541WktoiE2pSfGSzr0RFzq/nJoUb/RJUAAura58ztkP89SVfTFxPyy/PDk5OVkLFizQX//6V+3cuVPjxo3TyZMnPVcB1aSBCbFKH9FFMQ7vKa0YR7jSR3SpVdfAA6hYXfmcsx/mqSv7Ytp+WHrVz3lz58713PDtxhtv1Msvv6wePXpc9nX+nDXsD1Puxgeg+tSVzzn7YZ66si/VuR/+fH8bEVQqq7qCCgAAqD615vJkAACASyGoAAAAYxFUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGsvTXk6vq/E11i4uLLa4EAAD46vz3ti83x6/VQaWkpESSFBcXZ3ElAADAXyUlJXI4HJfsU6t/68flcunQoUOKjIyUzRbYH3wqLi5WXFyccnNz+R0hA/B+mIX3wyy8H+bhPbk0t9utkpISNWvWTCEhlz4LpVbPqISEhKhFixbVuo2GDRvyl8wgvB9m4f0wC++HeXhPLu5yMynncTItAAAwFkEFAAAYi6ByEXa7XampqbLb7VaXAvF+mIb3wyy8H+bhPQmcWn0yLQAAqNuYUQEAAMYiqAAAAGMRVAAAgLEIKgAAwFgElQq88soratmypcLDw9WjRw9t3rzZ6pKCVlpamrp3767IyEhFR0dryJAh+v77760uC5JmzJghm82myZMnW11KUDt48KBGjBihq6++WhEREerQoYO++eYbq8sKSk6nU3/+85/VqlUrRURE6Prrr9czzzzj0+/Z4OIIKr/w7rvvKjk5WampqcrIyFCnTp3061//WgUFBVaXFpRWr16t8ePHa+PGjVqxYoXOnDmjO+64QydPnrS6tKC2ZcsWvfbaa+rYsaPVpQS1n376Sb1791a9evX06aefKjs7W7NmzVKjRo2sLi0ozZw5U+np6Zo7d6527typmTNn6rnnntOcOXOsLq1W4/LkX+jRo4e6d++uuXPnSjr3e0JxcXF65JFHNHXqVIurw5EjRxQdHa3Vq1erb9++VpcTlE6cOKEuXbro1Vdf1f/93//pxhtv1EsvvWR1WUFp6tSp+vrrr7V27VqrS4Gku+++W02bNtXrr7/uabvvvvsUERGhxYsXW1hZ7caMygVOnz6trVu3asCAAZ62kJAQDRgwQBs2bLCwMpxXVFQkSWrcuLHFlQSv8ePH66677vL6nMAaS5cuVbdu3TR06FBFR0erc+fOWrBggdVlBa1evXpp5cqV2r17tyRp27ZtWrdunQYNGmRxZbVbrf5RwkA7evSonE6nmjZt6tXetGlT7dq1y6KqcJ7L5dLkyZPVu3dvJSQkWF1OUFqyZIkyMjK0ZcsWq0uBpP/85z9KT09XcnKyHn/8cW3ZskUTJ05UWFiYRo4caXV5QWfq1KkqLi5Wu3btFBoaKqfTqenTp2v48OFWl1arEVRQa4wfP15ZWVlat26d1aUEpdzcXE2aNEkrVqxQeHi41eVA58J7t27d9Oyzz0qSOnfurKysLM2bN4+gYoH33ntPb731lt5++221b99emZmZmjx5spo1a8b7UQUElQs0adJEoaGhOnz4sFf74cOHFRMTY1FVkKQJEybok08+0Zo1a9SiRQurywlKW7duVUFBgbp06eJpczqdWrNmjebOnauysjKFhoZaWGHwiY2NVXx8vFfbr371K/3zn/+0qKLg9thjj2nq1Kn67W9/K0nq0KGD9u/fr7S0NIJKFXCOygXCwsLUtWtXrVy50tPmcrm0cuVK9ezZ08LKgpfb7daECRP04Ycf6ssvv1SrVq2sLilo9e/fXzt27FBmZqbn0a1bNw0fPlyZmZmEFAv07t273OX6u3fv1rXXXmtRRcHt559/VkiI99dqaGioXC6XRRXVDcyo/EJycrJGjhypbt26KTExUS+99JJOnjypUaNGWV1aUBo/frzefvttffzxx4qMjFR+fr4kyeFwKCIiwuLqgktkZGS5c4MaNGigq6++mnOGLPLoo4+qV69eevbZZ/XAAw9o8+bNmj9/vubPn291aUEpKSlJ06dP1zXXXKP27dvr22+/1ezZszV69GirS6vd3Chnzpw57muuucYdFhbmTkxMdG/cuNHqkoKWpAofb775ptWlwe1233LLLe5JkyZZXUZQW7ZsmTshIcFtt9vd7dq1c8+fP9/qkoJWcXGxe9KkSe5rrrnGHR4e7r7uuuvcTzzxhLusrMzq0mo17qMCAACMxTkqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsfpQQgFH69eunjh07Kjw8XAsXLlRYWJh+//vf66mnnrK6NAAWYEYFgHH++te/qkGDBtq0aZOee+45Pf3001qxYoXVZQGwAD9KCMAo/fr1k9Pp1Nq1az1tiYmJuu222zRjxgwLKwNgBWZUABinY8eOXsuxsbEqKCiwqBoAViKoADBOvXr1vJZtNptcLpdF1QCwEkEFAAAYi6ACAACMRVABAADG4qofAABgLGZUAACAsQgqAADAWAQVAABgLIIKAAAwFkEFAAAYi6ACAACMRVABAADGIqgAAABjEVQAAICxCCoAAMBYBBUAAGAsggoAADDW/wd2prurS71ybAAAAABJRU5ErkJggg==",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#2b)\n",
        "n = np.arange(0, 10)\n",
        "x2 = n * np.heaviside(n, 1) - n * np.heaviside(n - 5, 1) + 3 * (n == 2)\n",
        "\n",
        "\n",
        "plt.stem(n, x2, use_line_collection=True)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('Amplitude')\n",
        "plt.title(r'$x_2[n] = nu[n] - nu[n-5] + 3\\delta[n-2]$')\n",
        "\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "b591eb92-e0e3-472e-bf9f-3c2df47ae3d4",
      "metadata": {
        "tags": [],
        "id": "b591eb92-e0e3-472e-bf9f-3c2df47ae3d4",
        "outputId": "27a11856-a274-4679-dcc4-f9fd748a4e1f"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "/tmp/ipykernel_56971/3998681574.py:5: MatplotlibDeprecationWarning: The 'use_line_collection' parameter of stem() was deprecated in Matplotlib 3.6 and will be removed two minor releases later. If any parameter follows 'use_line_collection', they should be passed as keyword, not positionally.\n",
            "  plt.stem(n, x3, use_line_collection=True)\n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "#2c)\n",
        "n = np.arange(0, 10)\n",
        "x3 = 2 * (n == 1) - 4 * (n == 3)\n",
        "\n",
        "plt.stem(n, x3, use_line_collection=True)\n",
        "plt.xlabel('n')\n",
        "plt.ylabel('Amplitude')\n",
        "plt.title(r'$x_3[n] = 2\\delta[n-1] - 4\\delta[n-3]$')\n",
        "\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "**3)** Resolvido por Catherine\n",
        "\n",
        "$$\n",
        "\\begin{aligned}\n",
        "x[n] &= 1 - \\sum_{k=2}^{\\infty} \\delta[n - 2 - k] \\\\\n",
        "&= 1 - \\sum_{k=2}^{\\infty} u[n - 2 - k] \\\\\n",
        "&= 1 - u[n - 2] - u[n - 3] - u[n - 4] - ... \\\\\n",
        "&= 1 - u[n - 2]\n",
        "\\end{aligned}\n",
        "$$\n",
        "\n",
        "Aqui, usamos a propriedade da função degrau unitário de ser igual a 1 para $n = 0$ e 0 para $n \\neq 0$.\n",
        "\n",
        "Para encontrar $n_0$, temos:\n",
        "\n",
        "$$\n",
        "x[n_0] = 1 - u[n_0 - 2] = 1\n",
        "$$\n",
        "\n",
        "Isso significa que $n_0 - 2 \\ge 0$, ou seja, $n_0 \\ge 2$. No entanto, como $n_0$ é o primeiro valor de $n$ para o qual $x[n]$ é diferente de 0, temos que $n_0 \\neq 2$. Portanto, $n_0 = \\boxed{0}$.\n",
        "\n",
        "Para encontrar $M$, temos:\n",
        "\n",
        "$$\n",
        "x[2M] = 1 - u[2M - 2] = 1\n",
        "$$\n",
        "\n",
        "Isso significa que $2M - 2 \\ge 0$, ou seja, $M \\ge 1$. Portanto, $M = \\boxed{2}$.\n",
        "\n",
        "Portanto, podemos escrever $x[n]$ como:\n",
        "\n",
        "$$\n",
        "x[n] = u[2n]\n",
        "$$\n",
        "\n",
        "Onde $u[2n]$ é a função degrau unitário que começa a ser não nula em $n = 0$ e tem um passo de 2."
      ],
      "metadata": {
        "id": "szlxtILNg-DU"
      },
      "id": "szlxtILNg-DU"
    },
    {
      "cell_type": "markdown",
      "source": [
        "**4)** Resolvido por Catherine\n",
        "\n",
        "$$\n",
        "x[n] = 0.5^{n-1}(u[n+2]- u[n-8])\n",
        "$$\n",
        "\n",
        "\n",
        "* **Para $n < 2$, $x[n] = 0$**\n",
        "\n",
        "Quando $n < 2$, a função degrau unitário $u[n+2]$ é 0, então a expressão $0.5^{n-1}(u[n+2]- u[n-8])$ também é 0. Portanto, $x[n] = 0$ para $n < 2$.\n",
        "\n",
        "Consequentemente, $x[n-k] = 0$ para qualquer valor de $k$, pois $x[n] = 0$ para $n < 2$. Isso significa que $a < 2$.\n",
        "\n",
        "* **Para $n > 8$, $x[n] = 0$**\n",
        "\n",
        "Da mesma forma, quando $n > 8$, a função degrau unitário $u[n-8]$ é 0, então a expressão $0.5^{n-1}(u[n+2]- u[n-8])$ também é 0. Portanto, $x[n] = 0$ para $n > 8$.\n",
        "\n",
        "Consequentemente, $x[n-k] = 0$ para qualquer valor de $k$, pois $x[n] = 0$ para $n > 8$. Isso significa que $b > 8$.\n",
        "\n",
        "* **Para $2 \\le n \\le 8$, $x[n] = 0.5^{n-1}$**\n",
        "\n",
        "Quando $2 \\le n \\le 8$, a função degrau unitário $u[n+2]$ é 1 e a função degrau unitário $u[n-8]$ é 0, então a expressão $0.5^{n-1}(u[n+2]- u[n-8])$ é igual a $0.5^{n-1}$. Portanto, $x[n] = 0.5^{n-1}$ para $2 \\le n \\le 8$.\n",
        "\n",
        "Consequentemente, $x[n-k] = 0.5^{n-k-1}$ para $2 \\le k \\le 8$, pois $x[n] = 0.5^{n-1}$ para $2 \\le n \\le 8$. Isso significa que $2 < a < 8$ e $2 < b < 8$.\n",
        "\n",
        "Portanto, os valores de $a$ e $b$ são:\n",
        "\n",
        "$\n",
        "a = 2\n",
        "$\n",
        "\n",
        "$\n",
        "b = 8\n",
        "$\n",
        "\n",
        "Logo,\n",
        "\n",
        "* Para $n < 2$, $x[n] = 0$, então $x[n-k] = 0$ para qualquer valor de $k$.\n",
        "* Para $n > 8$, $x[n] = 0$, então $x[n-k] = 0$ para qualquer valor de $k$.\n",
        "* Para $2 \\le n \\le 8$, $x[n] = 0.5^{n-1}$, então $x[n-k] = 0.5^{n-k-1}$ para $2 \\le k \\le 8$.\n",
        "\n",
        "Logo temos que a expressão para $x[n-k]$ é:\n",
        "\n",
        "$$x[n-k] = \\left\\{\\begin{array}{cc}0.5^{n-k-1}&\\text{ , se } 2< k < 8 \\\\ 0& \\text{ , c.c.} \\end{array}\\right.$$\n"
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      "source": [
        "**5)** Resolvido por Catherine\n",
        "\n",
        "a) $x_1[n] = \\left\\{\\begin{array}{cc} n & , 0 \\leq n \\leq 3\\\\n^2&, 3< n \\leq 7\\\\ 0& \\text{c.c.} \\end{array}\\right.$\n",
        "\n",
        "$x_1[n] = n \\cdot (u[n] - u[n - 3]) + n^2 \\cdot (u[n - 3] - u[n - 7])$\n",
        "\n",
        "\n",
        "\n",
        "b) $x_2[n] = \\left\\{\\begin{array}{cc} \\cos\\left(\\frac{\\pi}{6}n\\right) & , -3 \\leq n \\leq 3\\\\ 0&\\text{c.c.} \\end{array}\\right.$\n",
        "\n",
        "$x_2[n] = \\cos\\left(\\frac{\\pi}{6}n\\right) \\cdot (u[n + 3] - u[n - 3])$\n",
        "\n",
        "\n",
        "\n",
        "c) $x[n] = \\left\\{\\begin{array}{cc} 1 & , 0 \\leq n \\leq 2\\\\2&, 2< n \\leq 7\\\\ 1&7 < n \\leq 10 \\\\0&\\text{c.c.} \\end{array}\\right.$\n",
        "\n",
        "$x[n] = 1 \\cdot (u[n] - u[n - 2]) + 2 \\cdot (u[n - 2] - u[n - 7]) + 1 \\cdot (u[n - 7] - u[n - 10])$\n",
        "\n"
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