set_08.ipynb

{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": 1,
      "id": "c7e2e0d8",
      "metadata": {
        "id": "c7e2e0d8"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "6f53949b",
      "metadata": {
        "id": "6f53949b"
      },
      "source": [
        "# Q1: Metropolis-Hastings\n",
        "\n",
        "For the spatial part of Hydrogen atom, we can guess a wave function $\\Psi_{T,\\alpha} \\propto e^{-\\alpha r}$. So, $E_{Local}$ is equal to\n",
        "\n",
        "$$ E_L = \\frac{\\hat{H}\\,\\Psi_{T,\\alpha}}{\\Psi_{T,\\alpha}} = -\\frac{1}{2}\\left(\\alpha^2 - 2\\frac{\\alpha}{r} \\right) - \\frac{1}{r}\\,. $$\n",
        "\n",
        "Now we need to calculate the best-fit value for $\\alpha$. Note that there are 3 more complications here compared to the question from the last problem set.\n",
        "\n",
        "1. $r$ here is not Cartesian but is the spatial part of spherical coordinates.\n",
        "\n",
        "2. $E_L$ at $\\alpha = 1$ becomes independent of $r$ (Spoiler: this is infact the correct answer).\n",
        "\n",
        "3. For $\\alpha \\ne 1$ there's a singularity at $r = 1$ in which the potential diverges."
      ]
    },
    {
      "cell_type": "markdown",
      "id": "6aedd08f",
      "metadata": {
        "id": "6aedd08f"
      },
      "source": [
        "## (a)\n",
        "\n",
        "Like before, write a function to preform a Metropolis-Hastings algorithm to find a distribution for $r$ at a defined $\\alpha$ this time using a non-symmetric $Q(x^\\prime; x)$ as discussed in the lecture.\n",
        "\n",
        "P.S. Use two different functions for $Q$ at your choise."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "id": "06763320",
      "metadata": {
        "id": "06763320"
      },
      "outputs": [],
      "source": [
        "#metropolice\n",
        "def P(r, a = 0.5):\n",
        "    return np.exp(-a*r)\n",
        "\n",
        "def Q(r, r0 ,l= 1 ):\n",
        "    return np.exp(-l* (r-r0) **3)\n",
        "\n",
        "def Metropolis(N, P, Q, a):\n",
        "    m = []\n",
        "    for j in range(100):\n",
        "      x = np.random.uniform(0,1)\n",
        "      y = np.random.uniform(0,1)\n",
        "      z = np.random.uniform(0,1)\n",
        "      r = np.sqrt(x**2 + y**2 + z**2)\n",
        "\n",
        "      for i in range(N//100):\n",
        "          X = np.random.uniform(-1,1)\n",
        "          Y = np.random.uniform(-1,1)\n",
        "          Z = np.random.uniform(-1,1)\n",
        "          R = r +  np.sqrt(X**2 + Y**2 + Z**2)\n",
        "\n",
        "          w = (P(R, a) / (P(r, a)))\n",
        "\n",
        "          al = min(1, w)\n",
        "          s = np.random.rand()\n",
        "          if  s <= al:\n",
        "              m.append(R)\n",
        "          else:\n",
        "              m.append(r)\n",
        "    return m\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "n = 1000\n",
        "bins = 50\n",
        "r = np.linspace(0, 10, 100)\n",
        "m = Metropolis(n, P, Q , 0.7)"
      ],
      "metadata": {
        "id": "LJ-rUlcexV7f"
      },
      "id": "LJ-rUlcexV7f",
      "execution_count": 41,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plt.hist(m, bins = bins,  density = True)\n",
        "plt.plot(r, P(r, 0.5))\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "0OlxZrySxWMR",
        "outputId": "003b4b5a-cbf9-4183-cbda-2470b504478c"
      },
      "id": "0OlxZrySxWMR",
      "execution_count": 42,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "No handles with labels found to put in legend.\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "id": "522d5a0a",
      "metadata": {
        "id": "522d5a0a"
      },
      "source": [
        "## (b)\n",
        "\n",
        "Plot histograms for $r$ with different sizes for each choise of $Q$ and compare them to each other and the analytical function. Which choise of $Q$ is better? Do you think there's a mathematical reason behind this?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "id": "cf128b3e",
      "metadata": {
        "id": "cf128b3e"
      },
      "outputs": [],
      "source": [
        "# Your code here"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "44f3ab54",
      "metadata": {
        "id": "44f3ab54"
      },
      "source": [
        "## (c)\n",
        "\n",
        "As before, plot $\\langle E_L \\rangle$ and $\\sigma^2 = \\langle E_L^2 \\rangle - \\langle E_L \\rangle^2$ for different values of $\\alpha$ and show that indeed $\\alpha = 1$ is the best-fit value."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "id": "a8a832d1",
      "metadata": {
        "id": "a8a832d1"
      },
      "outputs": [],
      "source": [
        "# Your code here"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "76c17429",
      "metadata": {
        "id": "76c17429"
      },
      "source": [
        "# Q2: FFT (warm up)\n",
        "\n",
        "## (a)\n",
        "Using `'scipy.fft'`, calculate and the fourier modes of $x(t) = 2\\sin(t) + \\sin(10t)$. Plot the absolute values."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 43,
      "id": "b6f6c3e3",
      "metadata": {
        "id": "b6f6c3e3"
      },
      "outputs": [],
      "source": [
        "from scipy.fft import fft, fftshift, fftfreq"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 44,
      "id": "84a5ef9d",
      "metadata": {
        "id": "84a5ef9d"
      },
      "outputs": [],
      "source": [
        "n = 1000\n",
        "ti = 0    #s\n",
        "tf = 20   #s\n",
        "T = np.linspace(ti, tf, n)\n",
        "def x(T):\n",
        "    return (2*np.sin(T)) + (np.sin(10*T))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 45,
      "id": "f57fbca8",
      "metadata": {
        "id": "f57fbca8",
        "outputId": "4c379447-af11-42ff-9873-5a1ecad90930",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "plt.plot(T, x(T))\n",
        "plt.xlabel('T')\n",
        "plt.ylabel('x')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 46,
      "id": "31b287f9",
      "metadata": {
        "id": "31b287f9"
      },
      "outputs": [],
      "source": [
        "yf = fft(x(T))\n",
        "xf = fftfreq(n, d = T[1] - T[0])[ 0 : n//2 ]"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(xf , (2/n)*np.abs(yf[0 : n//2]), \".\")\n",
        "plt.xlim(0,3)\n",
        "plt.xlabel('f [Hz]')\n",
        "plt.ylabel('fft|x(T)|')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "GSYt7vGAzex8",
        "outputId": "515056fa-b132-4c43-8d4c-321308625114"
      },
      "id": "GSYt7vGAzex8",
      "execution_count": 47,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "id": "05dec31c",
      "metadata": {
        "id": "05dec31c"
      },
      "source": [
        "What's the prominent frequency? Does it make sense? If not, what's wrong with the code?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 48,
      "id": "a33fff66",
      "metadata": {
        "id": "a33fff66"
      },
      "outputs": [],
      "source": [
        "# Answer here"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "567fd6a9",
      "metadata": {
        "id": "567fd6a9"
      },
      "source": [
        "## (b)\n",
        "\n",
        "Calculate and plot Fourier transform of sawtooth function."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 49,
      "id": "89a9320a",
      "metadata": {
        "id": "89a9320a"
      },
      "outputs": [],
      "source": [
        "from scipy.signal import sawtooth"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 50,
      "id": "880d6f3c",
      "metadata": {
        "id": "880d6f3c",
        "outputId": "018702b7-24da-4498-b7d2-9e3456d66740",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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01Hp/p1NVikoiFAzuzd1XZma6gmSEcB9CdDpZgKouBBa2WfbjuNs/BH6Y5LkvAi/62kCT1PMfl2d01NGOhPlDFLQTzS3MX1LGjQX+REdN+4o/28+WPUf43/dfk/fTFYT9JLoJoUP1jby2fCdfudr/UUfz+OhAUrHoaDDnPsSKd5zYdAUjBpzDPROG+f56YT88ZgXEdNpLS6PR0SemBXflbZjnRAiSRUfdWhqpYW3FIR7L8HQF2cq2gOmUow1NvLR0O3de7m90NCbc+1/BCyQ6GifkO8CBKyrZxpC+PbkvQ9MVZDsrIKZTTkZH83DcH9daWpW5TqKj1vsDKN1Ry4ryWmbfdhE9u2d2uoJkbEZCkzPiRx29ekT/QF87vB+h4Px5417Kq+sC632Y0xWVRDi/dw8emDSy45XzhBUQk7LYqKOB9j7sexKInbzdRsHg3tx15YWBvW7YY6RB2VB5mEVbq/luhqcr6IgQ7p0nKyAmJdHoaDk3jB3I9Q6io/n+Jfbhlv18tvcoT0wdl/fRUReKSrbRt1d3vu3bdAXZyQqISclv11Sx90iDr6OOJmJXop+6cG3kwGCio/HsSBl8vu8o723ax0O3jKVPhqcryHZWQEyHmltaeWZxhGssOupELDr66BQ30dE87/wxtyQSna7g5jGBv7ZIuK/DsQJiOrRg3W4qao/zpMOTt2EekdRvFh11Z8eBOv5j3W6+deNoBth0BWewAmLa1dKqzC2JcOmFfZyMOprvh1BWO4iOxhMk1DFSv8WmK3h4sn+TpbUnehI9vNvfCohpV2zU0TnTHUdHw/sZ8lVRsUVHXanypit44PqRXNDH3+kKspUVEJNUkKOOJpPPHZANlYdZ/Hk1D08ONjoaL597gPMXlyECs32criDbWQExScVGHX3coqNOxKKj37rRbXQ0Hzt/+4828MbqCr42cYTT6QqiV6I7e/kOWQExCcWPOjoj4Ohowva4bkDAtu616KhLz3+0neaWVh6bar2P9lgBMQktKwvHqKP5OmTHvEURejuKjsbLxyvRD9Y18tqKndxzzTDG+DxdQcck1DtPTguIiNwlIltFJCIiTyV4/AcisllE1ovIhyIyOu6xFhFZ6/0saPtc0zVPF4crOppPX2Kx6OiDFh114qWl26lvbOHxAKcryFbOCoiIdAPmAncDlwMPiMjlbVb7FChU1auBt4F/jXvsuKpO8H7uCaTRecLFqKPJ5GMH5JlFZZztMDoaL996gEcamnh52Q7uuuJCLh7i/3QF2c5lD2QSEFHVclVtBN4EZsSvoKolqlrv3V0BhGN3OMcVlUQYaNFRJ2LR0Zkhio7m03Ugry6PTlcQ5GRp7bGT6MkNByri7ld6y5J5GPhT3P1eIlIqIitE5N5kTxKR2d56pdXV1V1rcR6IjTr6cMCjjnYkzBdTZdJzFh11pr6xmRc+3s7USwZz1Yh+rpuTFcLzDdEOEXkQKASmxC0erapVIlIAFIvIBlUta/tcVZ0PzAcoLCzMj2+hLphbEgnVqKP5dABl/9EG3lxdwV9f6zY62la+fGjeWFVBbV0jT4ZusrTw/gVc9kCqgPhjJCO8ZacRkTuAHwH3qOqJ2HJVrfL+LQcWARP9bGw++HzfUf68aS8P3TzGoqMOxKKjj1rvI3ANTS3MX1LGjQUDuW508NMVJBP2HSiXBWQ1MF5ExopID2AmcFqaSkQmAs8RLR7745YPEJGe3u1BwC3A5sBanqPmxUYdvcX9yduY2DncMB8HzoRwRUdPyZdz6G+vqWTfkROBT1eQ7ZwdwlLVZhGZA7wHdANeVNVNIvIToFRVFwD/CzgP+I2XBtnlJa4uA54TkVaiRfCnqmoFpAt2HKhjwbrdfG9ygUVHHYhFR8Ny8vY0OV68m1paeXZxGRNH9efmi8I1XUHYT6I7PQeiqguBhW2W/Tju9h1JnrcMuMrf1uUX16OOdiTEn6EuO9LQxEtedHS8RUcD9+7a3VQePM7/vOeKvIstd5VdiW6oOnScdz4N66ijuf+BfnX5To42NAc713yKcn1GyJZWZd6iCJcN7cv0S4OfriAVYe6BWAExzF9chqpFR12Ij45eOTyc0dEQf3912Z827qG8uo45DidLa0/YC7gVkDwXG3U0bNHRmFMn0XPza+xXK3eFNDoaFcLv1IxRVYqKI1w0uDd3XXmh6+ZkJSsgee4FG3XUmYamFn7xUTk3FZwfquhoW7lavD/csp/P9h7liWnhna5AJNwX0VoByWMH6xp5dcVO/ipk0dFEwvsRSl8sOhrGcx+5TlV5uiTCyIHncM817qcryFZWQPLYS8t2hDc66gnnfmHXNbW08syicEZH4+Xq9v84coB1FYd4bMo4ujucriAVYe4AhnvLGd8caWji5aXbbdRRR95du5uqQ8dDe/I2Xoi/v9JWVBzhwr69+Ovr2ht+z71wvzOsgOStsI06mszJL9cc+hZraVXmlYQ7OhoT8tqWltU7alm5vZbZtxU4n64g21kByUM26qhbf9q4h/IDdTw5Pfy9Dwj3IZR0FBVHOL93Dx6YNMp1UzokYjMSmpCJjTo6J+S9j3hhTqJ0RmtrXHT0CouOBm195SEWf17Nw5PHck4P6310lRWQPHOi+dSoo4VjwhsdjQn//nnnfPjZqejoWSGNjsbLhh5SZxQVR6cr+NaN4ZiuIBVh7gFaAckzNuqoO6pKURZGR3Ol97d171H+snkf37llrE1XkCFWQPJILDo6YWS4o6Pxcmk492yKjsbkUv9jbkmE3j268Z1bxrhuSs7IjnexyYgF3qij2XLyNtdkS3S0rVwo3tsP1PGH9bt58KbR9D83e6YrsCvRTSi0tCpzQz7qaHuy/Uts1XaLjrr0zKIIZ3c7i0duLXDdlJxiBSRPhH3U0WTCPhppqopKsic6epoc2PyVB+t555MqHpg0isF9erpuTk5xWkBE5C4R2SoiERF5KsHjPUXkLe/xlSIyJu6xH3rLt4rIF4Nsd7axUUfdWl95iCWfV/PI5IKsjI5meeeP+UvKEYHZt2Vf70OEUP8BnBUQEekGzAXuBi4HHhCRy9us9jBwUFXHAT8HfuY993Kic6hfAdwFzPN+n0kgNuro41PDO+poMrlwIXosOvrgjVnW+yD7e4D7jzTwpjddwbAQTleQ7Vz2QCYBEVUtV9VG4E1gRpt1ZgCveLffBm6X6PGXGcCbqnpCVbcDEe/3mTZOi45OyJ7oaK74bO+R7I+OZnH1/sVH5Vk/XUGYN39KBURE7heRPt7t/y4i74jItV187eFARdz9Sm9ZwnVUtRk4DJyf4nNjbZ8tIqUiUlpdXd3FJmefpZEa1nrR0bOzJDqaS+aVlFl01JHaukZeX7mLGROGM/r8cE9XkEzYe4CpfqP8D1U9KiK3AncALwDP+NeszFHV+apaqKqFgwcPdt2cwD1dvC0ro6NtZeOkRtkaHY2XRXmLM7y0dDv1jS08nsW9j7BLtYC0eP9+GZivqn8EuvqJqAJGxt0f4S1LuI6IdAf6ATUpPjfv2aijbuVKdDTM1yEkc6ShiZeX7eDuKy9kfBZPVyAS7p2nVAtIlYg8B3wDWCgiPTvx3GRWA+NFZKyI9CB6UnxBm3UWALO82/cBxRrdmguAmV5KaywwHljVxfbknGwadbQj4f0IJZYr0dFs7YC8unwnR7NguoJsl2oR+DrwHvBFVT0EDAT+risv7J3TmOP93i3Ar1V1k4j8RETu8VZ7AThfRCLAD4CnvOduAn4NbAb+DDyhqi1tXyOf5cqoo9l6COW5xdkbHW0rxDvACdU3NvP8R+VMu2QwVw7P/ukKwrz5u6eykqrWA+/E3d8D7Onqi6vqQmBhm2U/jrvdANyf5Ln/AvxLV9uQq+aWZN+oo7li/5EG3iqt4L7rLDrqwq9W7uJgfRNzcmDA0LDvP1ksJwdt3XuU9zbt46Fsjo56YimUbNoLjkVHH52S/Sdvs60H2NDUwvwl5dxUcD7XjR7gujk5zwpIDpq3yBt19OYxrpuSd2rrGnltRXZHR9vKotrNb9ZUsv/oCZ6cnhvnPkQk1DtPVkByzPYDdfzHumh0dEDv7IyOJhbiT1Gcl5Zup6E5d6KjYb8OIV5TSyvPLirj2lH9uSlLpivIdlZAckyuREdjsukQSiw6etcV2R0dbSvMMdJ4v/+0iqpDx5mTY9MVhHnrWwHJIVWHjudEdDRbWXTUnZZWZd6iMi4f2pdpl2TfdAXJhL0MWgHJIc8tLsuZ6GhMtsxImGvR0Zhs2ZFfuGEP2w/U2WRpAbMCkiNs1FG3cik62lbIazetrcrckgjjLjiPL16RY9MV5MiV6Cbknv94e9aPOtqe8H6ETkVHb77IoqMufPhZdLqCJ6ZdxFlZNl1BtrMCkgMO1jXy2oqd3HPNsJyJjsZkQwooFh2dk4PnPoRwHz6MTpa2jVEDz+Wvrs7N6QpCvPmtgOSC2KijdvI2eBYddeujbQdYV3mYx6ZeRPccnK4g7LtPubfF88yRhiZeyoFRR5MJ+0n0WHT0yenjc/Pkbcj/T0UlEYb268XXrs3u6QqylRWQLGfRUXdi0dErhvVl6iX5N9eMayvLa1iV49MViEioj2FZAcli9Y3NvPDx9pyLjiYSxjkpYtHROdMsOupCUUmEQef1YOb12T9dQbayApLFfrVyF7V1jTkZHY0J69dya6tSVJyj0dE4se0ftijpuopDfLTtAA/fWpDV0xWkIow7TzFWQLKUjTrq1gdb9rF1n0VHXSkqidDvnLN58Mbc7n2E/Z1lBSRLvZ1jo44mE8aT6KrRC9dyOToaE8Yjc5/tPcL7m/fxnVvGZP10BdnOSQERkYEi8r6IbPP+PWMXWkQmiMhyEdkkIutF5Btxj70sIttFZK33MyHY/4FbTS2tPLOojIkWHXUi16OjiYSpgM8tKaN3j248lCfTFYRp27fl6t3/FPChqo4HPvTut1UPfFtVrwDuAv6viPSPe/zvVHWC97PW/yaHx7trd3vRUTt560JRsUVHXSmvPsYf1u/mWzeNof+5uTRdQWJh/3i7KiAzgFe8268A97ZdQVU/V9Vt3u3dwH4g77OSLa3KvJJIzo06mpw3I6HjVsSsLK9h1Y5avp/D0dF4ErLt/8yiMnp2P4tHJo913ZTAWA/kTEO8edUB9gJD2ltZRCYBPYCyuMX/4h3a+rmIJB27XERmi0ipiJRWV1d3ueGuLdywh/IDdTk350G2OBkdnZTbJ2/DqKK2nt99WsXM60cx6Lz8mK4g7EP5+FZAROQDEdmY4GdG/HoazQcmrbEiMhR4FfiOqrZ6i38IXApcDwwE/j7Z81V1vqoWqmrh4MHZ3YGJH3X0rhyOjsY7dRLd/W5YLDr6yOQCep2d+70PCNchlOeWRKcr+P6U3JmuINt19+sXq+odyR4TkX0iMlRV93gFYn+S9foCfwR+pKor4n53rPdyQkReAv42g00Prdiooz//xjUWHXXgVHR0tOumBC5awN295/YdaeDXpZXcd90IhvbLr+kK7DqQMy0AZnm3ZwHvtl1BRHoAvwN+qapvt3lsqPevED1/stHX1oaAqlKUJ9HRMIqPjp7X07f9LpPEL5aU09KqPDYlt2PrbYWpB5iIqwLyU+ALIrINuMO7j4gUisjz3jpfB24DHkoQ131dRDYAG4BBwD8H2/zgfRw5wLqKQ3kVHYXwXEg1t6SM83p2z5voaMzJK9EdtqG2rpHXV+5ixjXDGHX+uQ5b4kYIjt4m5WRXSlVrgNsTLC8FHvFuvwa8luT5031tYAg9XRzhwr4WHXWhzIuOfv+2i/IiOho2L368nYbmFh6flpuTpWWz/NmVzWKrtteyanst35+SH9HReLGkmcu9sHyMjsa4PoRy+HgTr3jTFYy7IPemK8h2VkCygI066k5FbT2//7SKByblT3Q0EVcF/NXlOzh6Ir+nKwjxESwrIGG3ruIQSz6vzotRR8MoFh2dfZtFR4NWdyI6XcH0Sy/gimG5PV1BMmG/1ssKSMjNzZNRR5M5dRI3+P2wU9HRkXkXHY05eQjRwfZ/Y9UuDtY35XXvA8J9Et0KSIh9tvcIf7FRR505FR21k7dBa2hq4bkl5dx8UX5PVxDu/ocVkFCbl2ejjibiapyQ/bkAABStSURBVDj3fI+Ouvab0gqqj55gTo5PV5DtrICEVL6NOho2Fh09XZAFvKmllWcXl3Pd6AHcVGDTFYT5NLoVkJB6ZlEZPfI0OuqaRUfd+t2nVVQdOm5zzeM+Rt0RKyAhVHkw/0YdTebkcOIB7oT9cplFR2OC/gJraVWeWVTGFcP6MvWS7B78NFPsJLrplOcWl9uoo47UnWjmxaXbuT2Po6Mu/XHDHrYfqLPJ0jxh3wRWQEJm/5EG3iqtyMtRRxM5eRI9oNf71UovOmonb4Fg56NobVXmFkcYf8F53Hl5fkxXkO2sgITMLz7Kz1FHw6ChqYX5H5Vzy7jzuXZU/kZHEwniMMoHW/axdd9Rnpg2zqYriBPiI1hWQMKktq6R11ZYdNSVWHTUzn0EL366gq9cPdR1c0Ijb2ckNJ330lKLjibj94yEFh1NLKhj8Eu2HWB95WEez7PpClIRhtk4k7G/VEgcPt7Ey0stOurKyeionbxNyO+hTOYWRxjarxdfu3aEr6+TbcL+VnRSQERkoIi8LyLbvH8THnAWkZa4yaQWxC0fKyIrRSQiIm95sxdmNRt1NLEgPkAtrcq8kghXDu/L1IstOhoviO+vleU1rNpRy/dvK6BHd9unzSau/lpPAR+q6njgQ+9+IsdVdYL3c0/c8p8BP1fVccBB4GF/m+uv+kYbdbQjfu7//mH9bnbU1NuFa+3w8yjKyekKJuXngKEdCe8BLHcFZAbwinf7FaLzmqfEmwd9OhCbJ71Tzw+jk9FR630ErrVVmVdSZtFRR9ZWHOKjbQf43uQCep1t0xW0FfbdGVcFZIiq7vFu7wWGJFmvl4iUisgKEYkVifOBQ6ra7N2vBJLO8yois73fUVpdXZ2RxmeSjTraPvH5QpD3LTraLr87ZEXF0ekKvnnjaH9fKIuF+By6f3Oii8gHQKJduh/F31FVFZFkm2i0qlaJSAFQLCIbgMOdaYeqzgfmAxQWFobuT/GbNZVUHz3B/5s5wXVT8o6qMrckwujzLTraET8+OFv2HOGDLfv4r3dczHk9ffsqMj7y7a+mqncke0xE9onIUFXdIyJDgf1JfkeV92+5iCwCJgK/BfqLSHevFzICqMr4fyAATS2tPLuozKKj7fBzBzgWHf3p166y6GgSfl6HMLckwnk9u+f1dAUdCvk5OVefmgXALO/2LODdtiuIyAAR6endHgTcAmzWaCi6BLivvedng9/bqKMp8yNGWlS8zaKjKcr0tQhl1cf444Y9fOum0fQ71yZLa0/oDpvEcVVAfgp8QUS2AXd49xGRQhF53lvnMqBURNYRLRg/VdXN3mN/D/xARCJEz4m8EGjrM6ClVZlno446s7K8htU7DvLolIssOurAM4vK6Nn9LB6+1aYraE/YdyudHHhU1Rrg9gTLS4FHvNvLgKuSPL8cmORnG/0WG3X02Qevtd5HO/yakTAaHe3JN64fmdlfnGP8eGtW1EanK/j2TaPzfrqCbGe7Xg7YqKNunYqOjrXoaIoyWb+fXVxGNxFm32bTFaTChjIxp7FRR1Pnx0lci466s+9IA78preS+QpuuIBVhPzhhBSRgNupoejK1DxaLjn73lrEWHe2ETO0Ez19STosqj02xAUNzgRWQgH1ko446ZdFRd2qOneBXK3cxY8IwRg606QpSEfIOiBWQoBXZqKOdksmT6BYd7bxMBjxejE1XMNWG7MkVVkACZKOOujWvxKKjaetiAT98vIlfLtvJl64cyrgLzstMm/JEiM+hWwEJko066k5FbT2/X1vF30yy6GhnZKr/8ctlNl1BOjLRA2xuaWVnTV0GWnMmKyABWWejjqYl9vHp6pXoFh3tmq5s/7oTzbywdDu3X3oBlw/rm8FW5Yeuvvf/uGEP0/5tEWsrDmWoRadYAQlIUYlFR12Jj45e2K+X6+bknddX7uRQfRNPTLfeR2d1tf/R2hodMHTcBedx9fDMzzVkBSQAW/Yc4f3NFh1NSwZOolt0NH1dPYLS0NTCLz7azq3jBnHtKJuuIGh/2byPz/cd8+2aMysgAbDoqDs1x07w+sqdFh3tonQL+K9LK6g+esLOfXRButs+frqCL1/lzzVnVkB8ZtFRt15cup0Tza0WHU1TV/ZZG5tbeW5xOYWjB3BjwcCMtSmfdKUHuPjzajZU+XvNmRUQn9moo10TG8oknZ2ww/VNvGLR0YxIZ/vHpit4YrpNV9AV6fRAVJWi4gjD+vXiqxP9u+bMCoiPKmrr+f2nVTwwaZRFRx14ZfkOjll01InmllbmLYpw5fC+TL3YpitIV7pjwa3cXkvpzoN83+fpCqyA+Oi5JWWcZdHRLkl3x7XuRDMvLt3OHZdZdLQr0u05/HHDHnbU1DNn2njrfThQVBzMdAVWQHyy70gDv15to45mTCf78Sejo9b7yIjODCkei45ePOQ87rx8iI+tyg+dvQ7k010H+ThygNm3+T9dgZMCIiIDReR9Ednm/XtGvk9EponI2rifBhG513vsZRHZHvfYhOD/F+37hUVHnYmPjk606GiXpNN58Ds6mlfS2HxzSyL0P/dsvnmD/9ecueqBPAV8qKrjgQ+9+6dR1RJVnaCqE4DpQD3wl7hV/i72uKquDaTVKaqta+R1G3U0I05diZ46i45mXqrbPxYdHeNjdDTfdKbzvWn3YT7Ysp/v3jKW3gFcc+aqgMwAXvFuvwLc28H69wF/UtV6X1uVIS9+bKOOutLY3Mqzi8osOupILDr6mE1X4MS8kjL69OzOrICuOXP1Fx6iqnu823uBjg6UzgTeaLPsX0RkvYj8XESSRpxEZLaIlIpIaXV1dReanJrDx5t4ZdkOi45mSGdPwP7+0yp2H25gjkVHM6IzWzCo6Gg+6cz2j+w/xsKN3jVn5wRzzZlvBUREPhCRjQl+ZsSvp9Gzc0k7aSIyFLgKeC9u8Q+BS4HrgYHA3yd7vqrOV9VCVS0cPNj/OKGNOuqPVLrxsejoVcP7McWioxmVyvYPKjqab1I9gjVvUSTwa858O0imqncke0xE9onIUFXd4xWI/e38qq8Dv1PVprjfHeu9nBCRl4C/zUijuygWHbVRR92IRUefffA66304EFR0NJ+IpFa8K2rreXftbh66eQznB3jNmavdhAXALO/2LODddtZ9gDaHr7yig0S/Je4FNvrQxk771cpdHLRRRzPq5En0Dj5FFh31icRGAmh/+wcZHc07KRSQZxxNV+CqgPwU+IKIbAPu8O4jIoUi8nxsJREZA4wEFrd5/usisgHYAAwC/jmANreroamF+R+V26ijjlh01K0go6PmdHsPN/B2aSX3F45gSN9gpytwMra4qtYAtydYXgo8End/BzA8wXrT/WxfOn7jRUf/feZE103JKakcibLoqH9SKcWbdx/hgy37+cEXLg4kOppPokOZtN8FiU1X8KiDa87sTFcGNLW08qyNOuqr9j5CFh0NQDt/gLmLotMVzLppTGDNySftHT48cOwEv1q1k3snDHdyzZl92jLgdzbqqDMWHXUrsv8YCzfs4ds2XYEvOvo6eeFjb7qCaW5GvLAC0kUtrcq8Eht11C8nh3NPshMWi44+OtWio36IfYEl2we26Qr8l+y9f7i+iVeX7+RLVw3losFurjmzT1wX/WH9bht11KFYdPTrhRYdDVpFbT2/X1vF30waHWh01ES9vCw6XcEch9ecWQHpgtZWZV5JmUVH/dROTbboqP/am4/CVXQ0nyTbJz12crqCIVw21N01Z1ZAuuD9LfvYuu+oRUcDkKgXb9HR4LQ9jBIfHb2wX7DR0XyT6L3/+oqdHD7exBzH15xZAUlT7OStRUfdiEVHgxp11JzOZXQ0nyTqAcamK5g8fhATRvZ30KpTrICkacm2AxYdDcDJk7htdoHnLooEOupovjp1Ev3U9ncdHc03bd/7b62u4MCxE07PfcTYN1+aioq3WXTUkVh0NMhRR80pLzqOjuazxuZWnl1cxvVjBnBDwfmum2MFJB0ry2tYvcNGHQ1CojNLLkYdzVdtt//h+iZ+6Tg6mk/ankR/55NK9hxuYM708W4a1IZ9+6WhqMRGHXUlNuroN2+w6GiQYkdRXlnuPjqab2IHsJpbWnlmcRlXj+jHbeMHOW1TjBWQTlpbcYiPtll01BWLjrpzKjp6gdPoaL76w/o97Kyp54lp4RnxwgpIJxUVW3Q0SLEPiqrbUUfzVfyV6K+v2Mmh+iabLC1gqqemK7hkSB++cFl4rjmzAtIJ0ejoPouOOmLRUXfio6MTbbqCwP1l81627T/GE9PDdc2ZFZBOsOho8GIflQN1Fh11IXYdwq+96Kj1PoIlIijwdHGEsYN6h+6aM9uNTlFZdTQ6+tiUiyw66sBbqyssOurQqyt2RqOjY226gqBt2X2ExpZW/vW+q+kWot4HOOqBiMj9IrJJRFpFpLCd9e4Ska0iEhGRp+KWjxWRld7yt0Skh99tnldio466dKi+yaKjDtU3toTq5G0+aWxpZXj/c/jqxDPm1nPO1SGsjcDXgCXJVhCRbsBc4G7gcuABEbnce/hnwM9VdRxwEHjYz8baqKPuxH9fWXTUAW/7XzW8H1NsuoLAxd7+j04p4OwQjnjhpEWqukVVt3aw2iQgoqrlqtoIvAnMkOgu0HTgbW+9V4B7/WstPGvRUWdix+Bdjzqar87uFt3+c2yyNCd6nX0Wg/v05P6QTlcQ5nMgw4GKuPuVwA3A+cAhVW2OW560bycis4HZAKNGjUqrISMHnsvDk8faqKMOnNOjG/9875VMvcT2fl2447IhFP3NRJuuwJH/8ZXLaWxuDe01Z74VEBH5ALgwwUM/UtV3/XrdtlR1PjAfoLCwsP3Z6ZOw2KhbD95o19y40qfX2Xzl6mGum5G3RgwId+LQtwKiqnd08VdUAfH9thHeshqgv4h093ohseXGGGMCFL6zMqesBsZ7iasewExggUbHNi4B7vPWmwUE1qMxxhgT5SrG+1URqQRuAv4oIu95y4eJyEIAr3cxB3gP2AL8WlU3eb/i74EfiEiE6DmRF4L+PxhjTL6TtpOV5LLCwkItLS113QxjjMkqIrJGVc+4Zi/Mh7CMMcaEmBUQY4wxabECYowxJi1WQIwxxqQlr06ii0g1sDPNpw8CDmSwOZli7eoca1fnWLs6J1fbNVpVzxgOIq8KSFeISGmiFIJr1q7OsXZ1jrWrc/KtXXYIyxhjTFqsgBhjjEmLFZDUzXfdgCSsXZ1j7eoca1fn5FW77ByIMcaYtFgPxBhjTFqsgBhjjEmLFZA2ROQuEdkqIhEReSrB4z1F5C3v8ZUiMiaANo0UkRIR2Swim0TkPydYZ6qIHBaRtd7Pj/1ul/e6O0Rkg/eaZ4xUKVH/7m2v9SJybQBtuiRuO6wVkSMi8l/arBPI9hKRF0Vkv4hsjFs2UETeF5Ft3r8Dkjx3lrfONhGZFUC7/peIfOb9nX4nIv2TPLfdv7kP7fpHEamK+1t9Kclz2/3s+tCut+LatENE1iZ5rp/bK+F3Q2DvMVW1H+8H6AaUAQVAD2AdcHmbdR4HnvVuzwTeCqBdQ4Frvdt9gM8TtGsq8AcH22wHMKidx78E/AkQ4EZgpYO/6V6iF0IFvr2A24BrgY1xy/4VeMq7/RTwswTPGwiUe/8O8G4P8LlddwLdvds/S9SuVP7mPrTrH4G/TeHv3O5nN9PtavP4/wZ+7GB7JfxuCOo9Zj2Q000CIqparqqNwJvAjDbrzABe8W6/DdwuIuJno1R1j6p+4t0+SnR+lKTzwIfMDOCXGrWC6GySQwN8/duBMlVNdwSCLlHVJUBtm8Xx76FXgHsTPPWLwPuqWquqB4H3gbv8bJeq/kWj8/AArCA622egkmyvVKTy2fWlXd7n/+vAG5l6vVS1890QyHvMCsjphgMVcfcrOfOL+uQ63oftMNFJrQLhHTKbCKxM8PBNIrJORP4kIlcE1CQF/iIia0RkdoLHU9mmfppJ8g+2i+0FMERV93i39wJDEqzjert9l2jPMZGO/uZ+mOMdWnsxyeEYl9trMrBPVbcleTyQ7dXmuyGQ95gVkCwiIucBvwX+i6oeafPwJ0QP01wDPA38PqBm3aqq1wJ3A0+IyG0BvW6HJDoV8j3AbxI87Gp7nUajxxJClaUXkR8BzcDrSVYJ+m/+DHARMAHYQ/RwUZg8QPu9D9+3V3vfDX6+x6yAnK4KGBl3f4S3LOE6ItId6AfU+N0wETmb6BvkdVV9p+3jqnpEVY95txcCZ4vIIL/bpapV3r/7gd8RPZQQL5Vt6pe7gU9UdV/bB1xtL8++2GE879/9CdZxst1E5CHgK8A3vS+eM6TwN88oVd2nqi2q2gr8Isnrudpe3YGvAW8lW8fv7ZXkuyGQ95gVkNOtBsaLyFhv73UmsKDNOguAWFrhPqA42QctU7xjrC8AW1T1/yRZ58LYuRgRmUT0b+trYROR3iLSJ3ab6EnYjW1WWwB8W6JuBA7Hda39lnTP0MX2ihP/HpoFvJtgnfeAO0VkgHfI5k5vmW9E5C7gvwH3qGp9knVS+Ztnul3x58y+muT1Uvns+uEO4DNVrUz0oN/bq53vhmDeY34kA7L5h2hq6HOiiY4fect+QvRDBdCL6CGRCLAKKAigTbcS7YKuB9Z6P18CHgUe9daZA2wimj5ZAdwcQLsKvNdb5712bHvFt0uAud723AAUBvR37E20IPSLWxb49iJawPYATUSPMT9M9JzZh8A24ANgoLduIfB83HO/673PIsB3AmhXhOgx8dh7LJY2HAYsbO9v7nO7XvXeO+uJfjEObdsu7/4Zn10/2+Utfzn2nopbN8jtley7IZD3mA1lYowxJi12CMsYY0xarIAYY4xJixUQY4wxabECYowxJi1WQIwxxqSlu+sGGJOvRCQWtQS4EGgBqr37kzQ6ppMxoWUxXmNCQET+ETimqv/mui3GpMoOYRljjEmLFRBjjDFpsQJijDEmLVZAjDHGpMUKiDHGmLRYATHGGJMWi/EaY4xJi/VAjDHGpMUKiDHGmLRYATHGGJMWKyDGGGPSYgXEGGNMWqyAGGOMSYsVEGOMMWn5/572vbIjEX1yAAAAAElFTkSuQmCC\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "plt.plot(T, sawtooth(T))\n",
        "plt.xlabel('T')\n",
        "plt.ylabel('s')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 51,
      "id": "3378c2c3",
      "metadata": {
        "id": "3378c2c3"
      },
      "outputs": [],
      "source": [
        "yf = fft(sawtooth(T))\n",
        "xf = fftfreq(n, d = T[1] - T[0])[0 : n//2]"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(xf, (2/n)*np.abs(yf[ 0 :n//2]))\n",
        "plt.xlim(0, 10)\n",
        "plt.xlabel('f [Hz]')\n",
        "plt.ylabel('fft|s(T)|')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "SOmbO1Ia0le2",
        "outputId": "d9fa5970-fc9a-41b7-bb74-ae62c91fb1b2"
      },
      "id": "SOmbO1Ia0le2",
      "execution_count": 52,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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AZZEnGGM+NMZ02E9XAAO6gXTA9BhiCkbXIEAXyyml1GDFMyAKsO5nHVJhHzuYm4F3D/ZNEZkrIsUiUlxbWwvY6yDs38DlCi+U6/YHSXJbwaE9CKWUGpwvRJFaRL4DzAb+cLBzjDGPGWNmG2Nmjxw5Eui5kjq8UM7rD5KTlgxoDUIppQYrngFRCYyLeF5oH4siIhcAPwcuNcZ0D+QCQYNTg+i5UG5YehKgi+WUUmqw4hkQq4EiEZkkIsnANcC8yBNE5GTgUaxwqBnoBawhpnAPIhAqUvsCDEuzA0KHmJRSalDiFhDGGD9wK7AA2Aa8YozZIiK/EZFL7dP+AGQCr4rIehGZd5CX61UgahZT5EK5IMPSdYhJKaUOhSeeL26MmQ/M73HsnojHFxzK6wdMuAcROcTkjRpi0oBQSqnB+EIUqQcrdMMgsIaYghHrIHI1IJRS6pAkdEBEbrXhEiEYNASDBm8gSHqyB49LtAahlFKDlNgBYXCmubrte1KH9mFKSXKRluzWGoRSSg1SQgdEMKIHIWLt7hrahynF4yYtya2b9Sml1CAldkBE1CDcLmuIqTtgBUKKx0W69iCUUmrQEjogAhG3HA1t9x3qQSR7XKQmubUGoZRSg5TQARF5wyARIRgM7+Qa6kHoLCallBqchA6IqBsGuewahD80xOS2i9S61YZSSg1GYgdEz1lMQRPuQSS5SEvy0OnTW44qpdRgJHRARC6UE3uhnDdiiCkt2a2b9Sml1CAldED0vCe1NcQUUYM4SJE6tGeTUkqpg0v4gPjMEJOvZw0iOiD+ubaCM3//gdPTUEop1buEDojoWUx8pgeRlvzZhXJPLtvL/uYumjt9zrFNFc3c/tI6Z7M/pZRSCR4QDe0+slOtTfnc9l5M4YCwVlL7Agafvf3G9gMtbK5sAaCtO1ybWLKrlrfW76eh3XuYfwOllPriStiAaOnyUdfWzeSRGUDEXkwRs5iyU63dzEO9hdfXVDg/39YVDojQ93VKrFJKhSVsQJTWtQMwMc8KiNAsptA6iGS3i9wM66ZBTR1e/IEgb6zbT15mCgCt3eEhphY7INq7dVGdUkqFJGxA7LUDYnJeqAdB9BBTkotc+65yjR0+9jV0UNfWzTdOHAP03oNoj+hBVDV38shHu3XGk1LqqJWwAbGnth0RGD8iHYiY5hrai8kdDoiGdi/1bVZ9YWp+JhBdg2jpCvUgwsfmbzrA/7y3ndrW7vj/Mkop9QUU11uOxlNpfTsFw9JI8biB6CEmj0vwuF3kZlgF7KYOL0F7htKE4VaPIzIgwjWI8BBTKCzadS8npdRRKmEDYm9dO5Ps4SWwdm8FqzeQYj+OHGLyBeyAsHscrRFDTC2ddhhEhIYTEN1auFZKHZ0Sdohpb227U38AnLDYVNnihEV6sptkt4vGjvAQ0+icVJLdrl57EFEBYdcj9H4SSqmjVUIGhD9oaO32R/Ugpo/NAWDr/uaoYafcjCQa273Ut3eTk5ZEkttFZqrHKVIHg4bWUA0iaojJeqxTX5VSR6u4BoSIXCQiO0SkRETu6uX754jIWhHxi8hV/X3d0HYaEyMCYtzwNLJSPfgChpSk8K+Vm55MY4eP+nYvIzKtIafMFI/Tg2jz+gktoI4Mg1BvIrIHYYyhuSM8PVYppY5kcQsIEXEDDwMXA9OAa0VkWo/TyoDvAi8M5LVDi+Em52VGXs/pRYRqEADD0pNo6vBS39bNiIxwQIRqEC0RW25EroMIDTFFDjst2HKA0363OGqbDqWUOlLFswdxKlBijNljjPECLwGXRZ5gjCk1xmwEBrRzXnVLN7npSYwdlhp1/ISx2UC4YA3hHkRDu5cRGdYiucxUD232QrnmqICI7EGEhpjCobGvvoNOX0CnviqljgrxDIgCoDzieYV9bFBEZK6IFItIcTAY4PlbTsPjjm7+9IJQD8LtHMvNSLZqEG3hIaasiCGm0Awm6H2aa+Sx0M+EahZKKXUkS5gitTHmMWPMbGPM7GPG5DDN7i1Eml5gHUuJ6kEk0dTpo7HDGx5iiihSh3oQSW6JWkkdCobIukRoWCpyiqxSSh2p4rkOohIYF/G80D52yDyh28j1MCkvk7Qkd4+ASHa28R5h78MUWaQOraIenZNKR/dnewuRdYnQuS3ag1BKHQXiGRCrgSIRmYQVDNcA18XxerhdwhWnFFCYm+YcCy2WAxge0YNo6VGkHpOd1qMH4Y/6CuH9m7QHoZQ6GsQtIIwxfhG5FVgAuIEnjTFbROQ3QLExZp6I/AvwBpALfENEfm2MOeFQrvvbK2ZEPQ9ttwFE1SC8/iDd/gDNnT5EID87hc2VXYC1XUdo5XVkDSI8xKQ9CKXUkS+uW20YY+YD83scuyfi8Wqsoae4GRbRg3BmMaVYv3Z7d4CWTh9ZKR6yUj3OQrnIoaaoHkSvhW0/y0vquWDaqPj9EkopNQQSpkg9WJFDTM5COfsudG1dfpo7feSkJ5GR7KEjtHiul+muEO45RPYg3lq/n1ueKaa8oSN+v4RSSg2BIz4ghtsBIRIOi1APorXbR0uXn5y0JNJTPHT4AgSDJmpYqa9ZTNUt1rBUeaMGhFLqyJKwu7n2V1aqB5dYQ01ue/ZTln0r0lAPIjs1iYxkN8ZApy/g9CAykt3RNYgeM58A6tqsRXP7m7oOy++jlFKHyxHfg3C5hGHpyc4aCAj3INq6/bR0+pweBFhbbIR6DSOzUpyA6PYHnC0+WiJ6EKFdYquaOuP/yyil1GF0xAcEWIvlhkcGRGo4ICJ7EGAVqEOrqEdmpThTX9ui7h/RSw+iWXsQSqkjyxE/xARw3ZwJ5KSFp7tmhWoQXX5auuwidUQPIlSYHpmVwobyZudcsNZaRNYg6uwexP5D6EEsL6mjprWby08e9E4kSikVc0dFQNx81qSo56EeREO7ly5fkOxUDxnJ4amvoV5DflYq3kAQXyDo1CVGZ6dGzWIK9SCqmvsXEKv2NtDc6ePCiGmxjy7Zw87q1qiAWLqrlvKGTq6bM36gv65SSsXEUTHE1FNakhuXQGldO4Bdg7CGmHr2IMBaLBcqTI8dlkprt59g0NDlCzi9iaqIInVzh4/nV+7DF/jsJrX3vredX7+9JerYgeYualu7nftmAzy9vJQHFu2M1a+slFIDdlQGhIiQmeLhnY1ViMDpU0Y4PYhQDcIVMS22w+t3gmDssDSMsYKkod0aXpqcl0Frt98Jkbc2VPLzNzbz67e3YEz4Q98fCLJlfwsHmrvwR4TH/uZO/EFDY4fXOVbT2k1dW3fUeUopdTgdlQEBkJWahDcQ5GvTxzA1P4v05IgehNdPRrKHjFCvojvgFKnHDrP2eWrt8jvDSzMLrW3GQ72IXdVtADy3ooznVuxzrrm7tp1OXwB/0FBt31OirTscPtUt4ftM1LR0YwzUt4dD4/6FO7nmsU9j/E4opVTvjtqACE11vfX8qVHPO7r9tHf7yUgJ1yU6vQGn7hAKiJYunxMQoftQ7LfrELtqWjlx3DDmTBrOo0v2ONfcWNHkPK6wV14fiKhdVLdaARMMGue1QwvxAFbvbWB1aaMz3VYppeLpqA2Ik8cP49uzCzl+jHUPiXANIkC7N0B6ijuqVxEqUo/Nse5iZ/UgrL/uZxYOA8IzmUpq2jl2VCZfOnYkFY2dzj0nNlU2O9evaOy0fyYcALV2D6Khw4vfrkfURPQqyho6CAQNZbqth1LqMDhqA+L3V87k3qtOdJ4nu114XEJ7qAeR7HEWz4VqECkel3NPidaIHsS0sdm4XUJVUxeN7V7q2ropys9imh0+26paANhY0cxJ46wwCQXEgYj1E6HeQmQohHoVvkDQmSm1p7bN+X5tazc7DrT2+/e+//0dLN5WHXXsykeW88BCLYgrpaIdtQHRk4iQbm+t0dEdICPF7Syea+8O0NLlJys1ydmmo6XTT32bl/RkN5kpHkZlpbC/uZMS+8N7an6mc9e7rftb8AWCbK1qYfaEXPKzUqiw924KDUtlpXiosesSNa2RoRHayqOT0CSn3bXtzvd/O38bVz6yvNctyDu9AedmSWAF3UMflvDiqjLnWJcvwNqyRlaXNkT97LWPreCP7+8Y6NuolDqCaEBEyEjx0N5tDSdF9iA6vdb+TFmpHicgQj2IPLtHMXZYGuUNHZTUhAMiPyuVvMwUtla1sLO6Fa8/yIzCHApz06J6EHmZKYwdlhbuQdhBIQK1dlhEDitF9iDWljXS1u3n9TUVUb9Lc4ePc+/7MOpDfuv+FoIGdlaHf768oQNjYF99+PW7fAFW7q1nya66Q3k7lVIJTgMiQkaKx9mLKSPFQ3pSuAbR2uUjK9VDtr1VeEuX1YMIbSE+Z/Jwivc1smhrNWlJbgrsYva0sdls3d/CRztqATh5XC6FuelUNIV6EF2MyUklPzvFmdlUa3+dnJfh9CBCATF+eDp77PUbje1e54P9mU/3Ra2jeGDRTqpbulmyq9Y5trHCqoGUN3bQae8xFXqt/c2ddPnsY7XtVpAcaI3qgdzydDH3Leh/r6Kh3Rs1zbe5w8dXHviYNfsanWNef5BfvLnJWZOilPri0ICIkJHsprXLT5s9xBQqXHd4rWmumSkeUpPcJLtdziymUA/ipjMmkuRysXh7DVPyM3DZO8dOG5PNrppWnl5eyllT8xg/Ip3C3DSqmqy1EFVNnVZAZKVS69QgushO9TB+eLoz3FTW0EGy28UZU0Y4PYiNdtH76tnj2FPXztIS6y/+bVUtPPNpKVmpHrbub3EK7KFZVMbg9HT22h/MxuDc02JXjVXT6PQFnGBq7vCxeHs1z67Y169ZVCU1rZz634tYsOWAc2xVaQM7q9tYFFED2VTZzHMrynhjXfh25cYY/rFsb9RQW6xEbt+ulPp8GhARphfksHJvA82dXjKSPVGF69YuvzO8lJXqcdZBhAIiPyuVK2dZW2UU5Wc5rzltbDa+gKGmtZtbzra2/CjMTXfWQhxo7mLssDRGZadQY6+mrmntJj/bCo1QD6KioZPC3DSm5mfS2OGjod3LhvImROBnFx9HXmYyzywvBeCP7+8kOy2J314xg6CB9WVWMGysbGbyyAwAdlZbIbA3op4RCotdEUNQoQJ78b4GjIHmTh9LdoZ7JdsPtPD8yn08vmRPVB3k1TUV+IOGxdtqnGPry62ew+aI2Vyhx5FTgDdWNPPrt7fyxNK9ff9LO4i1ZY1c9tAnzgwygMXbqjnpNwupjNg3q6XLxz+W7dUFiUr1QgMiwve/NIVA0OALGNJTPFGF67ZuP5kp1vBSdloS9W3dNLR7ycsM7xL7b2dPxu0SZ/YS4Dwuys/kS8eMBKAw1xp+2nGghdZuP6NzUhmVnYo/aGjo8FoBkZXCqOwUZzV1WUMHhcPTnQ/4PbVtbChvYsrITIZnJHPtqeP5YEcNS3fVsnh7NTeeNoHzjsvHJbC6tIHWLh97atv5xsyxJLtd7LR7CXvr2zlmVCYApfVWQOysbqUwNw2XwHY7IFbtbSDJLeSmJ/HWhv2AVau4/vGV/PyNzfz3/G38Y1kpAIGg4U27R/DpnnrnvVgXCqqKZmfoaZMTEOFjn9g9oXc3VznHvP4gd/9zEx/tCAdOyOJt1Vz72IqoMHhhZRkbKppZGjHEtnh7DV5/kOUl4drKS6vK+PXbW1kaUW8xxlBS0/+ZYeUNHVHDZgDPflrKD55bE3Xsox01WvhXCUUDIsK44el8094wL9MeXkpP9tDhtbbRiOxBLNhSTdDAifYaCIDJIzNZ8JOzueH0Cc6xSXkZnF2Ux39edBwi1rBTKCBWl1ofKtYQk9UTqWnppqa1i/ysFPKzU53V1GUNHYwfnsbkPOvDfE9tOxsqmp3rXz9nAm4RfvjcWjwu4TunTSAzxcPxY7Ip3tfgfBCfMiGXySMznF7C3rp2Tho3jNz0JErrQ0NMbcwoyGHyyEy22VNoV+5t4MTCYXx95hgWbj1Ae7efdzZWUd/u5bEbZnHGlBG8vLqcQNCwfHcd1S3dzJk0nIrGTsrt9RsbypvISvHQ3OlzivShHkR9u9f5y36Z/QFe3tDJlv1WQL27uYoXV5Xxr0+t5tmI1enlDR385OX1fLqnnnnrrVDyB4LOMFZkb2f13gb7fZKGHMsAABdHSURBVA/P2ArVht7bHB4Ke3FVORfcv4RXi8s/899IaV07uyMmCRhjuPXFddzw95XONvEA/1heyrubD0TdivahD0r4ywclUT2YZ1fs47rHV/S6b1dvalu7PzNM9qPn1/KnHvt2Xf/ECn7+xqaoY+9truKTQ5x40OH1R9WVAN7fciAqnAH21bf3+3dSX1waED386LypdpE5HbAW0LXZM5tCAVGYm0ZWiodHrj+FCyJ2ZQWYmp9Fql3cBmt78GdvnhO1e+vYYWl4XMLj9irrMTlp5GdbC/CqW7qoaQkNMVmhsau6jeZOH+OHW/WL9GQ3f1y4g7q2bk4cZ63iHp2Tylenj6a1288lM8c6rzd7Qi7rypp4e0MVADMKcigalcXO6lZau3zUtnYzKS+TCSMyKK1rp8sXYF99O0Wjsjh+TDbbqlro8PrZXNnMqZOGc/lJBXT5gtz73naeXl5KUX4mF04bxXVzxlPZ1MmSXbU8t2If2akefnnJNACW766jpKaNdm+AK2cVAlaPocsXYFdNG2cX5QGwyT5WvK+Rb55SgNslzgf308tLmTginfOOzeeXb27mwx01dPkC3PriOjAwcUQ6r9ozuVbtbaCpw0duehJLd9VhjKGh3csuu+5SbAdzW7ef1aUNiMCibdUEggZjDE/bQ3W/eXurs/ixvKGDqx/9lHPv+4jLH1pGvb0GZumuOjaUN9HhDfCu3dbdtW3ssYfuPthu9Xjq27pZW2Zdd4F9ni8Q5KEPdrF8dz2vRITR8pI6/u8bm3h/y4Goek9JTRvn3/cRc58J90w2Vzbzv5uqePKTvc4kg7L6DpaV1PPGukrnWGuXjztf2cAdr6x3jgG8tqaCV4vLP/OhHwwaVu1toNsfPnd3bRuz/msRz68MT5PeWNHE3GfX8OdFu5xj5Q0dXHD/xzy4OHzMGMNTy/Y6ta++dPkCLN1VGzXxYsv+Zi5/eFnUzsnbqlq4+M9Lo4K4yxfgzXWVUW0fqJYuH9sPtEQd23GgNWqKOFiBGRqGjaW2br8zkSSkyxeI+nd3OGhA9DAxL4PiX1zA12aMBiAj2cOG8maMCd+q9HffnMknPzufi2eMGdQ1UpPcPHvzHK7+l3GcXZTHtLHZjMq2wqCkpo1uf9AeYrI+5EPDF+OHp+Nxu3j8xtmMy03H7RJOnzzCed25Z09meEYy/3b2ZOfY7InD6fAGeHFVGV8+Lp/hGckck59JRWP4r/NJeRlMystgX32HM4OpKD+T40ZnUdHYyZKdtfiDhlMnDWfWhFxuPmsST3+6j02Vzdx4xkREhK9MG82IjGRue2EdC7ZUc+PpEzlhbDZ5mSl8uruedfaH4zWnjiPJLWyqbGZbVQuBoOFbs61jGyqaKba3EvnGiWOZM2k48zdVsXJPPWvLmrjh9In89TuncMyoTO56fSO3vrCWjRVN/OFbJ3Lj6RPZWNHMjgOtvLflAKlJLm49v4iq5i5KatqcXsMFx49iT107ta3dLC+pwxcwXD9nPPXtXopLGyje18iO6lZ+dN4UAsbw4xfXUdvazQ+fX8vWqhZ+eO4UOnwB/mR/ID70QQmjs1MZPzzdmWq8cKvVe8nLTHEC4sMdtQQNZKd6eM8u3C/eVk11SzfDM5L586JddHoDNLR7ue3Fdbywsoy5z67hxidX4vUHaerwcsvTq2n3+vmkpI6V9tDd8yut3lRLl9+p98zfbP0x0OENOENyb6yrpMMboLa1m9fsdm7d38LPXt/IT1/byC/f2uz8xb+spI6vPbiUbz/6Kb98c7Pz39J//+82On0B/vphiRNcL622gu3N9ZXOsRdWleELGJ75dJ/T23lnYxW/ensrNz25yglXfyDIwx+W8Is3N/Hg4l3OB6I/EORHz6/lhr+v4tU14eC8b8EO1pc38feI2tSjH+9mW1ULf/8kfOyhD0r4ycvreWBhdGjd8fJ6Fmw5EBU6Xb6A04ZQHSoYNNz81Gq+8ZdPombX/fyNTdz9z01RNbQ7X97A1x9cGhUSH26v4faX1jm/Z4gvEHT+CIs89rPXNnJ/xNCj1x/ksoc+4ZrHVzhtDQQNV/1tOVc/+mlU+x/+sIT7F+6MCnhjTMyCJK73gxCRi4A/A27gCWPM73t8PwV4BpgF1ANXG2NK49mm/gjdPAisLb83VTZzwthszj02HyDq5kODdfqUEZw+JfzhnuQWROCdTVXOdfPt0Hh86R5E4LjRVj3jzKl5nDk1jw6vn/TkcFtPHDeMtb+8MOo6F04bxc8uOo45k4dzsr2Ku2iUVUQPFYEnj8xgwoh03lxf6fzHf8yoLGezwh+/uB63S5g1IRcR4RdfP55OX4AlO2udIblkj4trTh3HIx/t5hdfP56bz5qEiHDGlBF8vLOWyqZOctKSOHZUFseNzmZzZTMF9lDbKeOHcdzobDZWNGGMIcktnDpxOM0dPn7y8nqufmwFaUlurppVSIrHzX3fOpEr/rqcRdtq+NU3pnHR9NE0tHv53bvb+Pkbm9hd28Y5RSP56gmj+K93trJkVx1VTZ0ke1zcfNYkFm2rpri0gaUldWQku/k/XzmWV4oreH5lmdNTvPW8Io4fk80dL6/nnHs/pNMX4NEbZvHVE0bT2uXnhVVlVDV3saq0gV99YxrNnX4eWLSTisYOFm6tZnpBNqdOHMFzK60PyUVbqxmdncq3Zxfylw9LqG3t5rkVZYzNSeWBq0/i6sdWcNuL6wgEg7R0+XjntrNYX97EL97czPefW8OOA63Utnbz7M1z+MnL6/nTol08emM2b67bz7dmFbJ0Vx2vr63g6zPHMH9TFSeMzeZAcxfvbKziqyeM5rkV+5hRkIPLJTy6ZDffnj2OX761mZy0JC47aSz/WFbK1v0tnHdsPg8s2sn44elcPH00rxRXcMHxowgEDR9sr+HcY0fy0Y5a5m3Yz9dmjGbe+v2MH55OWUMHH+6wvv/y6nIm52Wwp66d19dU8K3Z4/j9u9uZOCKdquYufvDcWn535QweXLyLt9bvZ3hGMg3tXlaXNvC7b87ggYW7WLy9hlHZKfz+3e18Zdpoyhs7+HBHLdmpHl5aXc6PLyiiyxfgfzdVkexx8WpxOXd+5Rg6vQGe+GQP6cluHluym4unj2ZKfia3PF3MjupW3lhXyfSCbP5w1YkcaO7i129vcYZWM5I93P2143l+5T5WlzbiErh3wXb+ev0sVtt/PIT+f/zzNSezvrzJCftfv72FF//tNPbVd3Dbi+to6/aztqyRh687heNGZ/P7d7fz7IpSfAHDhBHpPHfzHAqGpfGz1zfyz7XW0OjJ43M577h8nluxz1kM+96WA3xtxhje3rCfzZVWCL21oZIrTi7kwx01/MGedp7kEm77chEHmrv47j9WUVLTxszCHOaeM5mLpo+JGhYdiLgFhIi4gYeBC4EKYLWIzDPGbI047Wag0RgzVUSuAf4HuDpebRqMP1w1k7ZuPxNGZMT1OikeN7d/ucj5yzS0yM4l4A8GeeT6U5iYF92GyHA4mNQkNz84d0rUsbOL8jixMIdF26oRsXomk/IyMMZaP5HkFiblZVCYm8aVpxSSnebhjCl5ZNlrQESE314xg0DQ4Lan8wLcccEx3HDaREbb+1UB3Hj6BD7dU8/q0kYunDYKEWF6QQ5vb9hPly9AbnoSBcPSmFmYw4uryli+u56zi/LISPFw+ckFTBiRzmtrKjhudJYTzDMLh3HvlTPp9AX4zmlWvWd4RjJXzRrHm+sqyc9O4cbTJ1KYaxX1n1+xD3/QcNK4YcyakEtqkotX11SwqbKZM6fmMSw9mfOOHck8u/h+y1mTSEt2c8nMsYzKTuXWF9byvTMn8tUTrF7lTy4o4u2N+1lb1si/nzOZ6+ZMoLqliwcW7eTmp4rZWdPKHRccwynjc3ly2V5eWlXOkl21XHFyARfPGMODH5Rw89Or2VjRzH9ceAxzJo/gx18u4u9L99DuDXDnhccwvSCH6QU51LZ28+fFuyjKz+TFuacxa0Iu3//SFP7rna2c9fsP6PQFuOmMieRlpfDYkj3M31TFxopm7rr4OMoaOnhzXSVPLS9lZ3Ub9145k2HpScx9dg3H3/MegaDhD1fN5Fuzx3HSuGH8/I3NrC1r4ivTRvGna07C43JR+vAy5j5rDWlNHJHO374zi8sfXsZDH+xyplA/fuNsbntxHS+vLqem1Zq88aerT+KPC3fyyEe7WbithsqmTl6aexrVLV38xysb+PIfPwbgp189lh+dN5VXisv5z9c2ctb/fAjAbedP5WszxnDJXz7h359bQ6c3QE5aEo/eMItrHlvBU8tK8dsTSh689iR+/OI6Hv6gxNmv7LUfnsEtTxfznSdWkpOeRFVzF099719oaPfy2/nb+fqDSwka64+jZ28+lfe3VPPokj1srWqhuLSRs4vyOGV8Ln9evIt3N1XxcnE5wzOSuXj6aF5aXc5t50/l3ve2MyIjmbnnTOZ3727n7n9uYl1ZE26X8Mj1p/DLt7Zw6UPLyMtMpq7Ny1WzCpk+NpsHFu3iir8uI8XjprKpk9vOn8r7W6r52esb+eUl03jwg12cOXUENS3d3Pf+Ds49diQPLNrJ8WOy8biE+xbsZHhGCj97bSPHjsri+DFZ/HHhTpaW1FHR0EFLl58bT5/Ikl21fP+5tRw7Kosd1f2fdBFJeo49xoqInA78yhjzVfv53QDGmN9FnLPAPudTEfEAB4CRpo9GzZ492xQXF8el3UPtrfWVvLSqnL/dMIuctCReX1PBcWOyOGFsTkyv4w8EeebTfTR1+rjzwmMoqWnjKw98zLGjs7n9y1O5aPrghs96Y4xhb107IzJTyElLYumuWm5/aT0N7V4unDaKx2+czfLddfz+3e1cePworj9tQtQ9xA/Fe5sP8Kt5WzjQ0sXtXy7ijguP4brHV7B8dz3ZqR7+dsMszpiSR2uXjx0HWhGBE8bmRNWRjDHOBIOQ5k4faUlukj3hUdoXVpbx+toK9tV38Or3T6dgWBqz/r+Fznbuz/zrqZxdlMc1j63gQEsXsycM555vTHOCr73bz4byJuZMHuEErzGGtWWNzCgY5lyr2x/giaV7qWzqZFxuOj84dwp7atu45C+f0GEP0yz56XlUNHZw3RMrAWsYcf6PzyY1ycW8DfvZfqCVjGQ3Pzx3qrNmp7yhg1V7G7j85ALn+nvr2nlh5T6m5mdy/nGjGJmVwofba/j359bg9QeZMjKDRXd+id+9u53H7JralJEZLLzjS3ywvYYfvbCWYelJXH5yAXdffDxgrfN5dU0FWakebjx9ovP+vbW+ku0HWrnylEKm5luTMR79eDePLtlDU4eXuy4+jrnnTOH6J1awrMQaYjvnmJE886+n8u2/fcoqexjxR+dN4adfPY7Nlc0882kpFY2dfGt2IVecbNW/Gtq93Pf+DiYMT+d7Z04i2ePC6w/y09c2sKu6jYLcNH516QkMS0viwvs/du43f+eFx3DVrELOufdDZzPNey6Zxk1nTOSmJ1exbHcdOWlJ3P/tEzn/uFE0d/h4dU05H++s5Zp/Gc/XZ1r/T20/0MI9b25hRGYyFxw/im+eUsCW/S1c/eintHsDuATeue1syho6+H7ETLi/3zSbtGQ31z1u/TtNTXLx+g/OoCg/i/sX7mTFnnq6fAHu+9aJTC/IwRcI8ujHu3l38wEunDaKOy48do0xZnY//rdxxDMgrgIuMsbcYj+/AZhjjLk14pzN9jkV9vPd9jmfmWohInOBuQDjx4+ftW/fvp6nqEPU0uUjy57eG2/GGCoaO8nNSHa2Wo8XfyDIuvImThibTXqyh9K6dvbWtXPG1BGkeNx9v8AhKKlppbyxk8wUD7PtIbp4qWvr5tPd9QSChstPLsAYwzsbqyjITWNmQQ4ed+xKjl2+AFv2tzAqO4XC3HQa2r28vqaCvKxkzpo60rkbY6xEhnRzp4+FW6vZ39TJ12aMYWp+JhWNHWwob2ba2GwmjkiP2fvc1OFl5d4Gyuo7uP608aQne3hjXQVl9Z3MKMzmvGPzEZFe/4gYqLZuP2X1HbhdwrGjszDGMG/DfvbVd5Cbkcx35oxHRFi+uw6XCEX5mc7mof0hIkduQEQ6knsQSikVD4MJiHjOYqoExkU8L7SP9XqOPcSUg1WsVkopNcTiGRCrgSIRmSQiycA1wLwe58wDbrIfXwV80Ff9QSml1OERt8FfY4xfRG4FFmBNc33SGLNFRH4DFBtj5gF/B54VkRKgAStElFJKfQHEtTpojJkPzO9x7J6Ix13At+LZBqWUUoOjK6mVUkr1SgNCKaVUrzQglFJK9UoDQimlVK/itlAunkSkFdA7r1jygEPb5P/IoO9DmL4XYfpehB1rjMnq+7Sw+O5xED87Broi8EglIsX6Xuj7EEnfizB9L8JEZMDbT+gQk1JKqV5pQCillOpVogbEY0PdgC8QfS8s+j6E6XsRpu9F2IDfi4QsUiullIq/RO1BKKWUijMNCKWUUr1KqIAQkYtEZIeIlIjIXUPdnqEiIuNE5EMR2SoiW0Tk9qFu01ATEbeIrBORd4a6LUNJRIaJyGsisl1Ettm3/j0qicgd9v8fm0XkRRFJ7funjgwi8qSI1Ng3ZQsdGy4iC0Vkl/01t6/XSZiAEBE38DBwMTANuFZEpg1tq4aMH/gPY8w04DTgR0fxexFyO7BtqBvxBfBn4D1jzHHAiRyl74mIFAA/BmYbY6Zj3XLgaLqdwFPART2O3QUsNsYUAYvt558rYQICOBUoMcbsMcZ4gZeAy4a4TUPCGFNljFlrP27F+hAoGNpWDR0RKQS+Djwx1G0ZSiKSA5yDdZ8VjDFeY0zT0LZqSHmANPtulenA/iFuz2FjjFmCdY+dSJcBT9uPnwYu7+t1EikgCoDyiOcVHMUfiiEiMhE4GVg5tC0ZUn8C/hMIDnVDhtgkoBb4hz3c9oSIZAx1o4aCMaYSuA8oA6qAZmPM+0PbqiE3yhhTZT8+AIzq6wcSKSBUDyKSCbwO/MQY0zLU7RkKInIJUGOMWTPUbfkC8ACnAI8YY04G2unHMMKRyB5fvwwrNMcCGSLynaFt1ReHfWvnPtc4JFJAVALjIp4X2seOSiKShBUOzxtj/jnU7RlCZwKXikgp1rDj+SLy3NA2achUABXGmFBv8jWswDgaXQDsNcbUGmN8wD+BM4a4TUOtWkTGANhfa/r6gUQKiNVAkYhMEpFkrILTvCFu05AQEcEaZ95mjLl/qNszlIwxdxtjCo0xE7H+m/jAGHNU/qVojDkAlIvIsfahLwNbh7BJQ6kMOE1E0u3/X77MUVqwjzAPuMl+fBPwVl8/kDC7uRpj/CJyK7AAa0bCk8aYLUPcrKFyJnADsElE1tvH/q99D3B1dLsNeN7+I2oP8L0hbs+QMMasFJHXgLVYs/7WcRRtuyEiLwLnAnkiUgH8P+D3wCsicjOwD/h2n6+jW20opZTqTSINMSmllDqMNCCUUkr1SgNCKaVUrzQglFJK9UoDQimlVK80IJRSSvVKA0KpgxCRH9tbZj/f4/h3RaRWRJ6wn5/bc5txEXlKRK76nNf+g4gcEJH/E5/WK3XoEmahnFJD4IfABcaYil6+97Ix5tbBvrAx5qci0j74pikVf9qDUKoXIvI3YDLwrojccQivM1tE1tv/bBIRXZmqEob2IJTqhTHm+yJyEXCeMaauHz9ydsS2JwDjgXeMMcXASWANKwHvxb61SsWHBoRSsbHUGHNJ6ImIPBX5TRG5Gmtn1a8c5nYpNWgaEErFmYhMB34FnGOMCQxxc5TqN61BKBVHIjIMeBG40RhTO9TtUWogtAehVHxdBkwAHrduSwDGmJOGtEVK9ZMGhFIHYd+EqD/nfQR81OPYdyOePo1SCUiHmJQauE7g4tBCucGwZzR9B+u+0Up9IekNg5RSSvVKexBKKaV6pQGhlFKqVxoQSimleqUBoZRSqlf/Px/6S5e+ZcZNAAAAAElFTkSuQmCC\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "id": "4fbb7db1",
      "metadata": {
        "id": "4fbb7db1"
      },
      "source": [
        "# Q3: Sunspot data"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "be22bf6d",
      "metadata": {
        "id": "be22bf6d"
      },
      "source": [
        "The attached file `Sunspots.csv` contains monthly solar activity data over a 272 year period. First read the data in a pandas Dataframe and tell pandas to interpret the second column as `datetime`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 53,
      "id": "42c142e4",
      "metadata": {
        "id": "42c142e4"
      },
      "outputs": [],
      "source": [
        "import pandas as pd"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 54,
      "id": "9c5b2072",
      "metadata": {
        "id": "9c5b2072",
        "outputId": "3ca8bf65-efbc-44a7-8740-d72c38401203",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 455
        }
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "            Date  Monthly\n",
              "Num                      \n",
              "0     1749-01-31     96.7\n",
              "1     1749-02-28    104.3\n",
              "2     1749-03-31    116.7\n",
              "3     1749-04-30     92.8\n",
              "4     1749-05-31    141.7\n",
              "...          ...      ...\n",
              "3260  2020-09-30      0.6\n",
              "3261  2020-10-31     14.4\n",
              "3262  2020-11-30     34.0\n",
              "3263  2020-12-31     21.8\n",
              "3264  2021-01-31     10.4\n",
              "\n",
              "[3265 rows x 2 columns]"
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-94272fe5-796c-4173-8d89-9eb91ff64f2e\">\n",
              "    <div class=\"colab-df-container\">\n",
              "      <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Date</th>\n",
              "      <th>Monthly</th>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>Num</th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>1749-01-31</td>\n",
              "      <td>96.7</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>1749-02-28</td>\n",
              "      <td>104.3</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>1749-03-31</td>\n",
              "      <td>116.7</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>1749-04-30</td>\n",
              "      <td>92.8</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>1749-05-31</td>\n",
              "      <td>141.7</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>...</th>\n",
              "      <td>...</td>\n",
              "      <td>...</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3260</th>\n",
              "      <td>2020-09-30</td>\n",
              "      <td>0.6</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3261</th>\n",
              "      <td>2020-10-31</td>\n",
              "      <td>14.4</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3262</th>\n",
              "      <td>2020-11-30</td>\n",
              "      <td>34.0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3263</th>\n",
              "      <td>2020-12-31</td>\n",
              "      <td>21.8</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3264</th>\n",
              "      <td>2021-01-31</td>\n",
              "      <td>10.4</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "<p>3265 rows × 2 columns</p>\n",
              "</div>\n",
              "      <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-94272fe5-796c-4173-8d89-9eb91ff64f2e')\"\n",
              "              title=\"Convert this dataframe to an interactive table.\"\n",
              "              style=\"display:none;\">\n",
              "        \n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "       width=\"24px\">\n",
              "    <path d=\"M0 0h24v24H0V0z\" fill=\"none\"/>\n",
              "    <path d=\"M18.56 5.44l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94zm-11 1L8.5 8.5l.94-2.06 2.06-.94-2.06-.94L8.5 2.5l-.94 2.06-2.06.94zm10 10l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94z\"/><path d=\"M17.41 7.96l-1.37-1.37c-.4-.4-.92-.59-1.43-.59-.52 0-1.04.2-1.43.59L10.3 9.45l-7.72 7.72c-.78.78-.78 2.05 0 2.83L4 21.41c.39.39.9.59 1.41.59.51 0 1.02-.2 1.41-.59l7.78-7.78 2.81-2.81c.8-.78.8-2.07 0-2.86zM5.41 20L4 18.59l7.72-7.72 1.47 1.35L5.41 20z\"/>\n",
              "  </svg>\n",
              "      </button>\n",
              "      \n",
              "  <style>\n",
              "    .colab-df-container {\n",
              "      display:flex;\n",
              "      flex-wrap:wrap;\n",
              "      gap: 12px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
              "      cursor: pointer;\n",
              "      display: none;\n",
              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "      <script>\n",
              "        const buttonEl =\n",
              "          document.querySelector('#df-94272fe5-796c-4173-8d89-9eb91ff64f2e button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-94272fe5-796c-4173-8d89-9eb91ff64f2e');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
              "          await google.colab.output.renderOutput(dataTable, element);\n",
              "          const docLink = document.createElement('div');\n",
              "          docLink.innerHTML = docLinkHtml;\n",
              "          element.appendChild(docLink);\n",
              "        }\n",
              "      </script>\n",
              "    </div>\n",
              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
          "execution_count": 54
        }
      ],
      "source": [
        "Sunspot = pd.read_csv('/content/Sunspots.csv', index_col=0)\n",
        "Sunspot"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 55,
      "id": "c3a63a41",
      "metadata": {
        "id": "c3a63a41",
        "outputId": "e604f16a-dcbd-4f7b-f6c1-9aeb692e1157",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "date = pd.to_datetime(Sunspot.Date)\n",
        "\n",
        "plt.plot(date, Sunspot['Monthly'], c = 'black')\n",
        "plt.xlabel('Date')\n",
        "plt.ylabel('Intensity')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "b98fe304",
      "metadata": {
        "id": "b98fe304"
      },
      "source": [
        "## (a)\n",
        "\n",
        "Do an appropriate Fourier analysis to determine the main cylce of solar activity. What's the period?"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "x = list(range(len(Sunspot.Date)))\n",
        "y = Sunspot.Monthly"
      ],
      "metadata": {
        "id": "sCIa2XWa7k5Y"
      },
      "id": "sCIa2XWa7k5Y",
      "execution_count": 56,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "\n",
        "#fast fourier transform and take absolute values\n",
        "f = abs(np.fft.fft(y))\n",
        "\n",
        "#get the list of frequencies\n",
        "num = np.size(x)\n",
        "freq = [i / num for i in list(range(num))]\n",
        "\n",
        "#get the list of spectrums\n",
        "spectrum = f.real*f.real + f.imag*f.imag\n",
        "nspectrum = spectrum/spectrum[0]\n",
        "\n",
        "#plot nspectrum per frequency\n",
        "plt.semilogy(freq,nspectrum, c = 'black')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "E9HAGe4ECdwF",
        "outputId": "ab7b6c61-4e93-41ec-a04d-799b89126125"
      },
      "id": "E9HAGe4ECdwF",
      "execution_count": 57,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f004c7b0850>]"
            ]
          },
          "metadata": {},
          "execution_count": 57
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#period\n",
        "results = pd.DataFrame({'freq': freq, 'nspectrum': nspectrum})\n",
        "results['period'] = results['freq'] / (1/52)\n",
        "plt.semilogy(results['period'], results['nspectrum'], c = 'black')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "EJ8O_bOxDR_r",
        "outputId": "53e24365-9f0c-41ab-edda-ba69f8ee3da8"
      },
      "id": "EJ8O_bOxDR_r",
      "execution_count": 58,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f0039db1390>]"
            ]
          },
          "metadata": {},
          "execution_count": 58
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 59,
      "id": "157ca964",
      "metadata": {
        "id": "157ca964",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "dedda013-fdfd-4fa6-c621-c071fa942155"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0               inf\n",
              "1       3265.000000\n",
              "2       1632.500000\n",
              "3       1088.333333\n",
              "4        816.250000\n",
              "           ...     \n",
              "3260       1.001534\n",
              "3261       1.001227\n",
              "3262       1.000920\n",
              "3263       1.000613\n",
              "3264       1.000306\n",
              "Name: freq, Length: 3265, dtype: float64"
            ]
          },
          "metadata": {},
          "execution_count": 59
        }
      ],
      "source": [
        "period = 1/results['freq'] \n",
        "period"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#period\n",
        "yf = fft(Sunspot.Monthly.to_numpy())\n",
        "xf = fftfreq(3265 , d= 30)[0:n//2]\n",
        "plt.plot(xf, (2/n)*np.abs(yf[0:n//2]),  c = 'black')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "uAMlCBO1HPAq",
        "outputId": "366590a6-2de1-462c-8094-16a1f7a56bd5"
      },
      "id": "uAMlCBO1HPAq",
      "execution_count": 60,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "id": "f86b4688",
      "metadata": {
        "id": "f86b4688"
      },
      "source": [
        "## (b)\n",
        "First do an inverse Fourier analysis to confirm that you get back the same data."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 61,
      "id": "8337e039",
      "metadata": {
        "id": "8337e039"
      },
      "outputs": [],
      "source": [
        "from scipy.fft import ifft"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "Sunspot.Date"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "qOCSh9HwPjDH",
        "outputId": "5fa44c72-1258-4f25-8dd5-ea096ac2ef6c"
      },
      "id": "qOCSh9HwPjDH",
      "execution_count": 62,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Num\n",
              "0       1749-01-31\n",
              "1       1749-02-28\n",
              "2       1749-03-31\n",
              "3       1749-04-30\n",
              "4       1749-05-31\n",
              "           ...    \n",
              "3260    2020-09-30\n",
              "3261    2020-10-31\n",
              "3262    2020-11-30\n",
              "3263    2020-12-31\n",
              "3264    2021-01-31\n",
              "Name: Date, Length: 3265, dtype: object"
            ]
          },
          "metadata": {},
          "execution_count": 62
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "Sunspot.Monthly"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "MNvdDifrPmQz",
        "outputId": "cf3d977a-b737-4759-d47f-e1de7062dad1"
      },
      "id": "MNvdDifrPmQz",
      "execution_count": 63,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Num\n",
              "0        96.7\n",
              "1       104.3\n",
              "2       116.7\n",
              "3        92.8\n",
              "4       141.7\n",
              "        ...  \n",
              "3260      0.6\n",
              "3261     14.4\n",
              "3262     34.0\n",
              "3263     21.8\n",
              "3264     10.4\n",
              "Name: Monthly, Length: 3265, dtype: float64"
            ]
          },
          "metadata": {},
          "execution_count": 63
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "fft(Sunspot.Monthly.to_numpy())"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "83Z_9vf-Qsq7",
        "outputId": "9a542912-6580-494c-b350-d78e5e4d7898"
      },
      "id": "83Z_9vf-Qsq7",
      "execution_count": 64,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([267007.7           -0.j        ,   7532.68481429+15868.49731631j,\n",
              "        -9485.36499832+16054.06182767j, ...,\n",
              "        -5146.14368204+38248.48932729j,  -9485.36499832-16054.06182767j,\n",
              "         7532.68481429-15868.49731631j])"
            ]
          },
          "metadata": {},
          "execution_count": 64
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(Sunspot.Date, Sunspot.Monthly, c = 'black')\n",
        "plt.plot(Sunspot.Date, abs(ifft(fft(Sunspot.Monthly.to_numpy()))), c = 'black')\n",
        "plt.show()"
      ],
      "metadata": {
        "id": "15-MusXXZfKJ",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "c217f395-2d27-4bf1-914f-f58640d45242"
      },
      "id": "15-MusXXZfKJ",
      "execution_count": 65,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#as you see we have same results."
      ],
      "metadata": {
        "id": "0R2DKaVjPz9N"
      },
      "id": "0R2DKaVjPz9N",
      "execution_count": 67,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "a58f33f8",
      "metadata": {
        "id": "a58f33f8"
      },
      "source": [
        "## (c)\n",
        "\n",
        "Remover the first ten biggest values of Fourier transformed series, do an inverse transform once more and plot the results."
      ]
    },
    {
      "cell_type": "markdown",
      "id": "c82dd951",
      "metadata": {
        "id": "c82dd951"
      },
      "source": [
        "Why does the plot look different? And what's the difference?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 68,
      "id": "aa3d2a38",
      "metadata": {
        "id": "aa3d2a38"
      },
      "outputs": [],
      "source": [
        "i = abs(yf)\n",
        "i.sort()\n",
        "i[-10:] = 0"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "j = ifft(i)"
      ],
      "metadata": {
        "id": "EBnKM8Xa7ww8"
      },
      "id": "EBnKM8Xa7ww8",
      "execution_count": 69,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(Sunspot.Date, Sunspot.Monthly)\n",
        "plt.plot(Sunspot.Date, abs(j))\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "BZCjdFTp7w96",
        "outputId": "b389c379-bfaf-4d20-a99b-f93cc6b9d72e"
      },
      "id": "BZCjdFTp7w96",
      "execution_count": 70,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#the first ten biggest values are the main values that we have! if we remove them its like we dont have any values so we see as a line!"
      ],
      "metadata": {
        "id": "h0bVVAV6ICQM"
      },
      "id": "h0bVVAV6ICQM",
      "execution_count": 71,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "dabdc648",
      "metadata": {
        "id": "dabdc648"
      },
      "source": [
        "# Q4: Correlation function and power spectrum"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "3f1c72a3",
      "metadata": {
        "id": "3f1c72a3"
      },
      "source": [
        "Remember from last homework that we calculated autocorrelation function for a data that we made?\n",
        "\n",
        "Load the data saved as `2pcf.txt`, calculate the correlation function once more and then plot the power spectrum using that data."
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.fftpack import rfft, irfft, rfftfreq, ifftshift"
      ],
      "metadata": {
        "id": "sRWUOUKRIZM3"
      },
      "id": "sRWUOUKRIZM3",
      "execution_count": 72,
      "outputs": []
    },
    {
      "cell_type": "code",
      "execution_count": 73,
      "id": "8ec4bc40",
      "metadata": {
        "id": "8ec4bc40",
        "outputId": "49dc1c40-eb80-4327-ae28-7a84ac94a0c4",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 424
        }
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "            0\n",
              "0      0.0000\n",
              "1     -0.5803\n",
              "2     -0.9645\n",
              "3     -0.7912\n",
              "4     -0.6154\n",
              "...       ...\n",
              "99995  1.8084\n",
              "99996  1.8084\n",
              "99997  2.2739\n",
              "99998  1.4093\n",
              "99999  1.4093\n",
              "\n",
              "[100000 rows x 1 columns]"
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-889d1da4-96d8-4d79-9d1c-383b74198357\">\n",
              "    <div class=\"colab-df-container\">\n",
              "      <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
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              "    .dataframe thead th {\n",
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              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>0</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>0.0000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>-0.5803</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>-0.9645</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>-0.7912</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>-0.6154</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>...</th>\n",
              "      <td>...</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>99995</th>\n",
              "      <td>1.8084</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>99996</th>\n",
              "      <td>1.8084</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>99997</th>\n",
              "      <td>2.2739</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>99998</th>\n",
              "      <td>1.4093</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>99999</th>\n",
              "      <td>1.4093</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "<p>100000 rows × 1 columns</p>\n",
              "</div>\n",
              "      <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-889d1da4-96d8-4d79-9d1c-383b74198357')\"\n",
              "              title=\"Convert this dataframe to an interactive table.\"\n",
              "              style=\"display:none;\">\n",
              "        \n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "       width=\"24px\">\n",
              "    <path d=\"M0 0h24v24H0V0z\" fill=\"none\"/>\n",
              "    <path d=\"M18.56 5.44l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94zm-11 1L8.5 8.5l.94-2.06 2.06-.94-2.06-.94L8.5 2.5l-.94 2.06-2.06.94zm10 10l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94z\"/><path d=\"M17.41 7.96l-1.37-1.37c-.4-.4-.92-.59-1.43-.59-.52 0-1.04.2-1.43.59L10.3 9.45l-7.72 7.72c-.78.78-.78 2.05 0 2.83L4 21.41c.39.39.9.59 1.41.59.51 0 1.02-.2 1.41-.59l7.78-7.78 2.81-2.81c.8-.78.8-2.07 0-2.86zM5.41 20L4 18.59l7.72-7.72 1.47 1.35L5.41 20z\"/>\n",
              "  </svg>\n",
              "      </button>\n",
              "      \n",
              "  <style>\n",
              "    .colab-df-container {\n",
              "      display:flex;\n",
              "      flex-wrap:wrap;\n",
              "      gap: 12px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
              "      cursor: pointer;\n",
              "      display: none;\n",
              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "      <script>\n",
              "        const buttonEl =\n",
              "          document.querySelector('#df-889d1da4-96d8-4d79-9d1c-383b74198357 button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-889d1da4-96d8-4d79-9d1c-383b74198357');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
              "          await google.colab.output.renderOutput(dataTable, element);\n",
              "          const docLink = document.createElement('div');\n",
              "          docLink.innerHTML = docLinkHtml;\n",
              "          element.appendChild(docLink);\n",
              "        }\n",
              "      </script>\n",
              "    </div>\n",
              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
          "execution_count": 73
        }
      ],
      "source": [
        "data = pd.read_csv('/content/2pcf.txt',header= None)\n",
        "data"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "data[0]"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "i1rkZuItGij4",
        "outputId": "caf08871-ba62-47ee-a70d-16ea2b5bd0a1"
      },
      "id": "i1rkZuItGij4",
      "execution_count": 74,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0        0.0000\n",
              "1       -0.5803\n",
              "2       -0.9645\n",
              "3       -0.7912\n",
              "4       -0.6154\n",
              "          ...  \n",
              "99995    1.8084\n",
              "99996    1.8084\n",
              "99997    2.2739\n",
              "99998    1.4093\n",
              "99999    1.4093\n",
              "Name: 0, Length: 100000, dtype: float64"
            ]
          },
          "metadata": {},
          "execution_count": 74
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "plt.hist(data[0], bins = 100, linewidth = 0)\n",
        "plt.show()"
      ],
      "metadata": {
        "id": "Gi7P2Ex8Lhi5",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "016efd6d-6dda-4082-ad33-af964dab0bbf"
      },
      "id": "Gi7P2Ex8Lhi5",
      "execution_count": 75,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "from statsmodels.graphics import tsaplots\n",
        "tsaplots.plot_acf(data, lags=100, use_vlines= False)\n",
        "#nemidunam chera 2 bar plot mikone!"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 600
        },
        "id": "ODCa-c3eNZD7",
        "outputId": "24c56a1b-1ba8-422d-fad4-dc6bce869246"
      },
      "id": "ODCa-c3eNZD7",
      "execution_count": 76,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.\n",
            "  import pandas.util.testing as tm\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {},
          "execution_count": 76
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#autocorrelation using fourier\n",
        "def autocorrelation(x):\n",
        "    xp = ifftshift((x - np.average(x))/np.std(x))\n",
        "    n, = xp.shape\n",
        "    xp = np.r_[xp[:n//2], np.zeros_like(xp), xp[n//2:]]\n",
        "    f = fft(xp)\n",
        "    p = np.absolute(f)**2\n",
        "    pi = ifft(p)\n",
        "    return np.real(pi)[:n//2]/(np.arange(n//2)[::-1]+n//2)\n",
        "\n",
        "u = autocorrelation(data[0].to_numpy())\n",
        "plt.plot(x[:100], u[:100], '.')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "PRLsKbP0AykY",
        "outputId": "599f45aa-15b3-45b6-c6c4-35f68aabdfc6"
      },
      "id": "PRLsKbP0AykY",
      "execution_count": 77,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#result is same as last result that we saw."
      ],
      "metadata": {
        "id": "y9yp3sebA7cP"
      },
      "id": "y9yp3sebA7cP",
      "execution_count": 78,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "#power spectrum\n",
        "power = fft(j)\n",
        "f = fftfreq(j.size)\n",
        "f = fftshift(f)\n",
        "power_  = fftshift(power)"
      ],
      "metadata": {
        "id": "pJeOVO-pJRV4"
      },
      "id": "pJeOVO-pJRV4",
      "execution_count": 79,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(f, abs(power))\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 267
        },
        "id": "GmSlC5dmJVep",
        "outputId": "181072a8-cf5f-4c9f-e2fd-92fd7a157960"
      },
      "id": "GmSlC5dmJVep",
      "execution_count": 80,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 81,
      "id": "09b0705a",
      "metadata": {
        "id": "09b0705a",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "outputId": "efda156b-9f67-4628-fc65-6610b32a23fe"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f0038470990>]"
            ]
          },
          "metadata": {},
          "execution_count": 81
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "y = data[0].to_numpy()**2\n",
        "yf = rfft(y)\n",
        "xf = rfftfreq(len(data[0]))\n",
        "plt.plot(xf, np.abs(yf))"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "e535676a",
      "metadata": {
        "id": "e535676a"
      },
      "source": [
        "# Q5: Real world example (Bonous question)\n",
        "\n",
        "The two point correlation function is like autocorrelation but for multi-dimensional data. Here instead of \"lags\" we use distances.\n",
        "\n",
        "The file `cmb_camb_N1024_PS12.npy` is part of the CMB temperature map, you can read it directly using `np.load` function.\n",
        "\n",
        "Calculate the two point correlation for the CMB map, plot it, and then calculate the power spectrum and plot it too.\n",
        "\n",
        "P.S. the provided data is 1024$\\times$1024, it needs too much computational power to calculate correlation for every point. You may use only a fraction of data points, if they are chosen randomly, the result would be the same. Also for the distances, you do not need to cover the whole length of the image."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 83,
      "id": "1609d819",
      "metadata": {
        "id": "1609d819",
        "outputId": "f064fb6a-0882-4fec-99c8-f127fdd3c7c4",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[-58.86565019, -52.00479425, -45.52710632, ..., -81.6929155 ,\n",
              "        -73.78002855, -66.13692367],\n",
              "       [-55.94739134, -49.27693447, -43.07373188, ..., -78.91851077,\n",
              "        -70.82681261, -63.13741826],\n",
              "       [-54.47494619, -48.02817536, -42.09378046, ..., -77.28168017,\n",
              "        -69.1433192 , -61.51249752],\n",
              "       ...,\n",
              "       [-75.58483339, -68.70942337, -61.74952149, ..., -95.43718643,\n",
              "        -89.00184571, -82.36334943],\n",
              "       [-68.92469854, -61.93094363, -55.0374856 , ..., -90.22468918,\n",
              "        -83.139877  , -76.01144799],\n",
              "       [-63.24349284, -56.26327424, -49.5423045 , ..., -85.53139564,\n",
              "        -77.95155842, -70.48945155]])"
            ]
          },
          "metadata": {},
          "execution_count": 83
        }
      ],
      "source": [
        "CMB = np.load('/content/cmb_camb_N1024_PS12.npy')\n",
        "CMB"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "07adc3ef",
      "metadata": {
        "id": "07adc3ef"
      },
      "outputs": [],
      "source": [
        "# Your answers here"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3.9.5 64-bit",
      "language": "python",
      "name": "python39564bit2ac12f2adc2c4da097ebd7138b514b17"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.10.4"
    },
    "colab": {
      "name": "set_08.ipynb",
      "provenance": [],
      "collapsed_sections": []
    }
  },
  "nbformat": 4,
  "nbformat_minor": 5
}