iFFT,  Radian frequency, with no 2pi, explicit.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Тестирование инварианта для обратного преобразования Фурье, частоты в радианах."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat{f(\\xi)} = \\int_{-\\infty}^{\\infty}e^{ix\\xi}f(x)dx$,\n",
    "\n",
    "инвариант: $f(\\xi) = e^{-\\frac{\\xi^2}{2}}$, константа: $\\sqrt{2\\pi}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Обоснование:\n",
    "\n",
    "Преобразование Фурье вида $\\hat{f(\\xi)} = \\int_{-\\infty}^{\\infty}e^{ix\\xi}f(x)dx$, действуя на $f(x) = e^{-\\frac{\\xi^2}{2}}$, принимает вид\n",
    "\n",
    "$\\hat{f(\\xi)} = \n",
    "\\int_{-\\infty}^{\\infty}e^{ix\\xi} e^{-\\frac{x^2}{2}}dx = \n",
    "\\int_{-\\infty}^{\\infty}e^{ix\\xi -\\frac{x^2}{2}}dx = \n",
    "\\int_{-\\infty}^{\\infty}e^{-(\\frac{x}{\\sqrt{2}})^2 + ix\\xi - (\\frac{i\\xi}{\\sqrt{2}})^2 + (\\frac{i\\xi}{\\sqrt{2}})^2}dx = \n",
    "\\int_{-\\infty}^{\\infty}e^{-(\\frac{x}{\\sqrt{2}} - \\frac{i\\xi}{\\sqrt{2}})^2 - (\\frac{\\xi}{\\sqrt{2}})^2}dx = \n",
    "e^{-(\\frac{\\xi}{\\sqrt{2}})^2}\\int_{-\\infty}^{\\infty}e^{-(\\frac{x}{\\sqrt{2}} - \\frac{i\\xi}{\\sqrt{2}})^2}dx =\n",
    "\\color{blue}{// t = \\frac{x}{\\sqrt{2}} - \\frac{i\\xi}{\\sqrt{2}}, dt = \\frac{1}{\\sqrt{2}}dx//}=\n",
    "e^{-\\frac{\\xi^2}{2}} \\cdot \\sqrt{2\\pi}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Библиотечное преобразование имеет неизменный вид $\\hat{f(\\xi)} = \\int_{-\\infty}^{\\infty}e^{-2\\pi ix\\xi}f(x)dx$\n",
    "\n",
    "То, которое нас интересует - $\\hat{f(\\xi)} = \\int_{-\\infty}^{\\infty}e^{-ix\\xi}f(x)dx$.\n",
    "\n",
    "Для того, чтобы их связать, используем явный вывод для связи дискретных формул.\n",
    "\n",
    "$$  a_k = \\frac{1}{M}\\sum_{l=0}^{M-1}A_l\\exp\\left\\{2\\pi i{kl\\over M}\\right\\} \\qquad k = 0,\\ldots,M-1.$$\n",
    "\n",
    "$a_k =$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import fft, pi, sqrt\n",
    "from scipy import integrate\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "def make_rad_ifft(f_hat_xi, d):\n",
    "    '''\n",
    "    Вычисляет обратное преобразование Фурье без двух \"пи\" в показателе экспоненты\n",
    "    \n",
    "    Параметры:\n",
    "    \n",
    "    f_xi - массив значений функции, задаваемый \"как есть\", без особенностей хранения\n",
    "    и определённый на диапазоне [-pi/d, pi/d] в M = 2**N равноудалённых точках, без последней.\n",
    "    \n",
    "    d = dx - желаемый шаг по переменной x, после того, как ifft отработает\n",
    "    \n",
    "    Области определения:\n",
    "    \n",
    "    xi_space = np.linspace( -pi/dx, pi/dx, num = M, endpoint = False)\n",
    "    x_space = np.linspace( -M*dx/2, M*dx/2, num = M, endpoint = False)\n",
    "\n",
    "    '''\n",
    "    dx = d\n",
    "    M = len(f_hat_xi)\n",
    "\n",
    "    sign_change_k = np.array([(-1)**k for k in range(0, M)])\n",
    "    sign_change_l = np.array([(-1)**l for l in range(0, M)])\n",
    "\n",
    "    f =  (M/(dx)) * sign_change_k * fft.ifft(fft.ifftshift(sign_change_l * f_hat_xi))\n",
    "    return f\n",
    "\n",
    "#old\n",
    "# def make_rad_ifft(f_hat_xi, d):\n",
    "#     '''\n",
    "#     Вычисляет обратное преобразование Фурье без двух \"пи\" в показателе экспоненты\n",
    "    \n",
    "#     Параметры:\n",
    "    \n",
    "#     f_xi - массив значений функции, задаваемый \"как есть\", без особенностей хранения\n",
    "#     и определённый на диапазоне [-pi/d, pi/d] в M = 2**N равноудалённых точках, без последней.\n",
    "    \n",
    "#     d = dx - желаемый шаг по переменной x, после того, как ifft отработает\n",
    "    \n",
    "#     Области определения:\n",
    "    \n",
    "#     xi_space = np.linspace( -pi/dx, pi/dx, num = M, endpoint = False)\n",
    "#     x_space = np.linspace( -M*dx/2, M*dx/2, num = M, endpoint = False)\n",
    "\n",
    "#     '''\n",
    "#     dx = d\n",
    "#     M = len(f_hat_xi)\n",
    "    \n",
    "\n",
    "#     sign_change_k = np.array([(-1)**k for k in range(0, M)])        \n",
    "#     sign_change_l = np.array([(-1)**l for l in range(0, M)])\n",
    "\n",
    "#     f =  1/np.sqrt(2*np.pi) * ((2*pi)/(dx)) * sign_change_k * fft.ifft(sign_change_l * f_hat_xi)\n",
    "#     return f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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TT08ydnCM/r39rS5JTeD7rLXEUCA1aGp2atFprQ6+z1pLDAVSg7rXdy86rdXB91lrySmt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\nbeTlI+sAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f44c51b4cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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t89ct2qyWZsWZmzdbR6HR0e+WUkl1GlsNPRukWpqdQGr7qEaJoUIaIJ4NUm0W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\nkiR1hKFCkiR1hKFCkiR1xP8PHVn1F8DQYA0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f44c4cd9a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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OSPcgWKeGSlpF7v0htYX7d0jqM0OF1BYuLCapzwwVUlu4sJikPqtsTIWkVebC\nYpL6zFAhtYULi0nqM7s/JElSKQwVkiSpFIYKSZJUCkOFJEkqhaFCkiSVwlAhSZJKYaiQtDJ79sBJ\nJ8HRRxc/77mn3xVJ6hNDhaSVcc8RSR2GCkkr454jkjoMFZJWxj1HJHW4TLeklXHPEUkdhgpJK+Oe\nI5I67P6QJEmlMFRIkqRSGCokSVIpDBWSJKkUhgpJklQKQ4UkSSqFoUKSJJXCUCFJkkphqJAkSaUw\nVEiSpFIYKlRL4+Pj/S5BJfL9bBffT/VSWaiIiKdExMcjYioi9kbE5RFxyALXHBIRH4yIb0XEgxHx\n5Yh4c1U1qr78n1a7+H62i++neqmypeIq4DhgE/AS4IXAZQtcczHwYuAU4NjO4w9GxEsrrFOSJJWg\nklAREccCm4E3ZOYXMvNG4Gzg1RGxbp5LTwSuyMzPZ+Y3M/Ny4IvAc6uoU5IklaeqlooTgb2ZeWvX\nseuABJ43z3U3Ai+LiKcBRMSLgGOA7RXVKUmSSnJARa+7Drin+0Bm7ouI+zvP9XI28CfAXRHxKLAP\neGNm3jDPNQcB3H777SurWLUyNTXF5ORkv8tQSXw/28X3sz26PjsPKuP1lhQqIuIC4Nx5TkmKcRTL\n9TaKloyXAt+kGIdxaUR8JzP/b49rjgI49dRTV3Bb1dHIyEi/S1CJfD/bxfezdY6i6C1YkcjMxZ8c\ncRhw2AKn7QJeC1yYmY+dGxFrgIeBV2XmNXO89kHAFPDLmfnXXcc/Ajw9M39xnpo2A9/ovL4kSVqc\ngygCxfbMvG+lL7aklorODRe8aUTcBBwaEcd3javYBARwc4/LDuz82Tfr+D7mGfvRqemqhWqSJElz\nWnELxbRKBmpm5k6KwZUfiYgTImIj8AFgPDN3T58XETsj4uWda/4F+Fvgwoj4mYg4KiJeD7wO2FZF\nnZIkqTxVDdSEYq2JD1LM+tgP/DlwzqxzjgGGuh6fDFwAfAx4KvDPwG9m5p9UWKckSSrBksZUSJIk\n9eLeH5IkqRSGCkmSVIpGh4qI+K2IuCEiftBZWGuuc46MiE93ztkdEe+JiEb/vQdJRHwjIvZ3/dkX\nEe/od11anIg4KyLujIiHImJHRJzQ75q0PBFx3qzfxf0R8U/9rkuLExEviIhrI+LbnffuZXOcc35E\nfKezoednI+I/LPU+Tf9wPRC4GvjjuZ7shIe/ohiQ+nzgNOD1wPmrVJ9WLoHfAdZSrMa6nmImkWou\nIk4G3gfozE3PAAADk0lEQVScBxxPsY/P9og4vK+FaSX+kR/+Lq4DTupvOVqCQ4DbgLdQ/H91hog4\nF3gr8CaK/bZ+QPH7+sSl3KQVAzUj4jTg4sx86qzjvwBcC6zPzHs7x94M/AHwo5n56KoXqyWJiDsp\n3ttL+l2LliYidgA3Z+Y5nccBfAu4JDPf09fitGQRcR7w8swc7nctWpmI2E+x0OS1Xce+A7w3My/u\nPP4RYA9wWmZevdjXbnpLxUKeD/zDdKDo2E4xjfU/9ackLcM7I+LeiJiMiLd3VmdVjUXEgcAI8Lnp\nY1l8g7mOYsNBNdMxnebzOyLiYxFxZL8L0spFxI9TtDx1/75+j2KxyiX9vla5TkUdrKNIWt32dD33\nxdUtR8vwfmASuB/4aYpWpnXA2/tZlBZ0OLCGuX//nrX65agEOyi6j79C0Q35u8D1EfETmfmDPtal\nlVtH0SUy1+/rfJuAPk7tWioi4oI5BgPNHqj3zH7XqeVbynucmX+Umddn5j92FkH7DeDszjdhSask\nM7dn5l90fhc/C/wi8BTgV/tcmmqkji0VFwIfXeCcXYt8rd3A7NHma7ueU3+s5D2+heLf7VHA10qs\nSeW6l2LfnrWzjq/F371WyMypiPgqsOQZAqqd3RR7c61lZmvFWuDWOa/ooXahYrGbli3STcBvRcTh\nXeMqXkyxG6pTofpkhe/x8RTLvt9TXkUqW2Y+EhETFBsJXguPDdTcBDjotgUi4skUgeLKfteilcnM\nOyNiN8Xv55fgsYGazwM+tJTXql2oWIrOIKGnAj8GrImIZ3ee+nqnj+8zFOHhzzrTZdYDvwd8MDMf\n6UfNWryIeD7FP+q/Af6FYkzFRcCfZeZUP2vTolwEbO2Ei1uAMeBgYGs/i9LyRMR7gU9S7Mn0dOBd\nwCPAeD/r0uJExCEUITA6hzZ0PjPvz8xvAX8E/E5EfB34BsVn5V3ANUu6T5OnlEbERyl2MZ3tRZl5\nfeecIynWsfhZinm3Wyk2Kdu/SmVqmSLieOBSioF9TwLupPhWdLGhsBki4i3AOyiaUW8Dzs7ML/S3\nKi1HRIwDLwAOA74L/B3w25l5Z18L06JExM9QfEGb/aF/RWae0TnndynWqTgU+DxwVmZ+fUn3aXKo\nkCRJ9VG72R+SJKmZDBWSJKkUhgpJklQKQ4UkSSqFoUKSJJXCUCFJkkphqJAkSaUwVEiSpFIYKiRJ\nUikMFZIkqRSGCkmSVIr/D71MoPR4shbUAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f44c4f77e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M = 2**10\n",
    "dx = 20 / M\n",
    "\n",
    "x_space = np.linspace( -M*dx/2, M*dx/2, num = M, endpoint = False)\n",
    "xi_space = np.linspace( -pi/dx, pi/dx, num = M, endpoint = False)\n",
    "\n",
    "#инвариант для преобразований без двух пи\n",
    "f0 = np.exp(-(xi_space * xi_space)/2)\n",
    "\n",
    "plt.plot(xi_space[(xi_space>-5) & (xi_space<5)], f0[(xi_space>-5) & (xi_space<5)], 'g.')\n",
    "plt.show()\n",
    "\n",
    "f1 = 1/np.sqrt(2*np.pi) * make_rad_ifft(f0, dx)\n",
    "\n",
    "# Рисуем только те же значения xi, для которых мы определяли функцию по x, \n",
    "# чтобы сравнение картинок получилось наглядным\n",
    "# Мы не использовали сдвиг ответа, но это то, что нужно сделать, если потом нужно применять не ifft\n",
    "\n",
    "plt.plot(x_space[(x_space>-5) & (x_space<5)], f1.real[(x_space>-5) & (x_space<5)], 'b.')\n",
    "plt.show()\n",
    "\n",
    "# мнимая часть мала, но есть. Можно раскомментировать, чтобы на неё посмотреть.\n",
    "plt.plot(xi_space[(xi_space>-10) & (xi_space<10)], f1.imag[(xi_space>-10) & (xi_space<10)], 'r.')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [conda root]",
   "language": "python",
   "name": "conda-root-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}