{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Product of independent complex-circular Gaussian Random Variable"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $z\\sim \\mathcal{CN}(0, \\sigma^2)$ and if $z = x + {\\rm j}y$, both $x$ and $y$ are zero mean Gaussian r.v. with variance $\\sigma^2/2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# magic\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f2c3e50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# prettyplot stuff\n",
    "import seaborn as sns\n",
    "sns.set(style='ticks', palette='Set2')\n",
    "sns.despine()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": [
       "iVBORw0KGgoAAAANSUhEUgAAAeQAAAFbCAYAAAAEMv1ZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n",
       "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3X+MXNd5H/zvnZ2d4ezOhkNKVKiX1EqE0lypJAOybEAx\n",
       "scqQQkX2dd6+oei2roi2VmOASgiIrFKENl3Y8ouKNvFaCRmUqSvjdR0kogVFoVoUAiilWlelDFko\n",
       "CBLZCPJpIywtkRZjhvayXO5wh7Nz3z9mz/DOnXPuPffHzL0z8/0AArjz4865M6N57jnnOc+xHMcB\n",
       "ERERpSuXdgOIiIiIAZmIiCgTGJCJiIgygAGZiIgoAwYiINu2/VzabUjKIJ0LMFjnw3PJJp5LNvFc\n",
       "whuIgAzgK2k3IEGDdC7AYJ0PzyWbeC7ZxHMJKR/lSbZtjwD4FoBfBOAAeFoI8X6SDSMiIhomUXvI\n",
       "vw6gIYT4FIB/A+D55JpEREQ0fCIFZCHEfwawf+nPBwD8LKkGERERDSMrTqUu27a/A2APgM8IIf48\n",
       "5HOLAH4ZwCcAFiM3omkGwLqYx8iKQToXYLDOh+eSTTyXbOK5dBoBcC+A/yGEWPDeGSsgA4Bt2z8P\n",
       "4D0ADwshqprHPIfBmuAnIiJK0lcjBWTbtv8ZgLVCiK/Ztv1zAC6gGZA7Ir7PMR4E8FcvvfQSVq9e\n",
       "HboNbn/5l3+JDRs2xDpGVgzSuQCDdT48l2ziuWQTz6XTlStXsG/fPgD4BSHEh977I2VZA3gVwHds\n",
       "234bwCiAg2GC8ZJFAFi9ejXWrl0bsRlNf/3Xfx37GFkxSOcCDNb58FyyieeSTTwXX8pp2kgBeWlo\n",
       "+p/Eag4RERG1DEphECIior7GgExERJQBDMhEREQZwIBMRESUAQzIREREGcCATERElAEMyERERBnA\n",
       "gExERJQBDMhEREQZwIBMRESUAQzIREREGcCATERElAEMyERERBnAgExERJQBDMhEREQZwIBMRESU\n",
       "AQzIREREGcCATERElAEMyERERBnAgExERJQBDMhEREQZwIBMRESUAQzIREREGcCATERElAEMyERE\n",
       "RBnAgExERJQBDMhEREQZwIBMRESUAQzIREREGcCATERElAEMyERERBnAgExERJQBDMhEREQZwIBM\n",
       "RESUAQzIREREGcCATERElAEMyERERBnAgExERJQBDMhEREQZwIBMRESUAfm0G0A0CI5PT+GHs1cA\n",
       "AA9VVuPQxp0pt4iI+g17yEQxHZ+ewgezV+AAcAB8MHsFh997DR/N/TTtphFRH2FAJopJ9ozdZmtV\n",
       "nHz/7RRaQ0T9KtKQtW3bowC+DeB+AEUA/1YI8V+SbBhR2rzD0I9iecotIqJBFrWHvA/AVSHE3wOw\n",
       "G8C/S65JROlTDUO/VP1QOQz9UGV1x22VQgkH1m/vfkOJaGBEDch/CuDLrmPUk2kOUTaohqFvOnXl\n",
       "MPShjTtRKZRaf1cKJRzbugeT5ZVdbSMRDZZIAVkIcVMIMWfb9gSawflLyTaLqL8cWL8dlUKJPWMi\n",
       "isxyHCfSE23bvg/AaQAnhRDfCXjscwC+orrvxIkTWLVqVaQ2EHXL67c+xuXGfNtt41Yeu4prcHdu\n",
       "WUqtIqJ+dvXqVRw8eFB391cjBWTbtn8ewH8D8NtCiO9FaZht2w8AmHnrrbewdu3aKIdoOXfuHLZs\n",
       "2RLrGFkxSOcC9Pf5HH7vNczWqgCaw9D/OD/Zt+fi1c+fixfPJZt4Lp0uXbqExx57DADWCSEueu+P\n",
       "Ood8BMByAF+2bft7S/+x20ADhcPQRNRLkZY9CSEOAtD2u4kGwWR5JY5t3dP6+ypmUmwNEQ06FgYh\n",
       "IiLKAAZkIiKiDGBAJiIiygAGZCIiogxgQCYiIsoABmQiIqIMYEAmIiLKgEjrkIkoPO92joc27ky5\n",
       "RUSUJewhE/WAajvHw++9ptzOkYiGEwMyUQ+otnOcrVWV2zkS0XBiQCYiIsoAziET9cBDldX4wNNL\n",
       "HsRNKzhPThQde8hEPXBo405UCqXW35VCCce27sFkeWWKrUoW58mJ4mFAJuqRQd/OkfPkRPFwyJqo\n",
       "R7zbOaaNw8t6fG8oDewhE3XB8ekpPH32FJ4+ewrHp6fSbk6H1299nPjw8kOV1R239eNoAIfeKS0M\n",
       "yEQJ64cf9MuN+Y7b4g4vD8o8OYfeKS0csiZKmN8Pepgh634cNj2wfnsrcMXtGffj+RPFwYBMFEAG\n",
       "BgfAw9PXexIYZC9bkr3sA+u3J9LjXJMb6+glRxleVgXNJObJu33+foZliRplDwMykYs3wAAIHRiS\n",
       "+EFPqpet8+ll9+GV+keYrVVb7dMdV9dTjRs0/XrA3T5/P4c27sTh914zem+IksQ5ZMqUNJOhVHO/\n",
       "3sAKBM8n9stcqlyGlbdymK1Vle+533x4nLnWuPPs7u/J67c+NnpOGIO+RI2yiQGZMiPtZChVgIkq\n",
       "7g96LzKWJ8srce/YctSdBgD1e96NBCdvz1p1XL/z935PLjfmE/+eyCVqWbyQosHFgEyZ0S/ZrSaB\n",
       "Me4PethedpSRBZPAqDN3eyHSRYPuNb38zr9fvidEYTEgEy3RBZiJ0WLb373qNZn2sqOMLMh1yEFU\n",
       "7wkA1J0GPpm/Hvq98RuF8J4nh41p2DAgU2akXVhC1yt7ZsMOVAoljFv5ngYG0152lB6jah2y5H7P\n",
       "ve+J9zUcB4kETQtoO8/j01M4ev4MrtequHdsedv5p/09IeoWBmTKjCwkQ6l6ZTIw7is9mKn5RDlM\n",
       "7SR4TG9gBPzXE+dzuVBD86pgagF4yt7W+juox+/9noxbec710kBgQKZMSXuYsl+SedxBSyXo/VuT\n",
       "G+u4zRsYpcnySjycUK9UddH1zUefxNZ71rVuM+nxu78nu4prQrWBKKu4DpkyJWsbMGRV0Fzssa17\n",
       "fNf5hlmH7PcaUSRRzcv9PTl37lykYxBlDQMy0QCxgLalQZKqaIdpYFRlRtcbDXw099NIowhBF11p\n",
       "VMpimU7KAg5ZE3VZN4qd6BKbjmzebbw0KE7S2Fx9oWvLjHqdS5D2+nciiT1kogQlUXrTRDfLO7rP\n",
       "YSxfSDRpzFSSm1QESbNMJ5EbAzJRQlTDxCpJ/dj7BS2TYV/VMK33HG7Wa9rXrxQ6E8NUogwHM5eA\n",
       "hhGHrIkSkmTpTRN+Q86q9cPXa1WcnrkA4E5hEO8wrUmxEOni3LXAod1+GA7mumbKCgZkoh7r1Y/9\n",
       "gfXbkbfu/C/uDoiqwiBy+DuMoAIk/VDmMgvr34kADlkTJUY3TLzoNHDj9kLrb+9QbLcyfCfLK7G4\n",
       "tHGEm1/gzVu51mYTw6SXc9ZEOgzIRDG5A6o7oMng+9HcTzt+7OVzvAlTSSV9BcnBQkORrrV2fAVm\n",
       "a/PGveWg3n4aS5ii4Jw1ZQEDMg29OD1UbxJU3WnAAlAeLXaU3tQ9xyuppK/j01PKDOlKoYSduXsw\n",
       "1fhJR+C9OHcNI5alPaYFtI5pktndzWxwokHDOWQaanGTjlRzpA6AESun7eH2IvlLF/QnRos4tnUP\n",
       "7s4t0/ZSFx39QqfyaBGVQgl5K4fZWtV3bbVcfz1bqyJv5RLtGbvXdr9+6+NEjkmUNvaQaahlcQ1q\n",
       "mMClWvesGgqX3LF2sryyrcdr2q7TMxdawd59EeMeZleNHCTFe+zLjfmeDPMTdRsDMlEMUeZIVc+R\n",
       "8lbO+ELAdN1z2/Fz7YNifm1xk7tAAfqLmOfPn4GFZjER1frlOBc67gsP1QWE37FZFpP6BQMyDbW4\n",
       "SUdh5kh1yV9udaeBw++9hhXFMVy8cQ1AM8DNLwU4d0AJO/StOq9DG3di/9lTvs/T7QKl4sC/mEgU\n",
       "QXPuYZ7bq6Q5oig4h0xDLYk1qCZbRnrnqmXyl8psrYqZG9daj71Zr8UurKE7r6Da2qrtEVWFNEzb\n",
       "EGUO2eTCQ3fsMOugu1FznCgMBmQaenH3YParmCV/5FU9vKg1omVAMQmMFpqJXLrz8gt2uuepqoCZ\n",
       "tKNbxTbGrXzsY/dDRTEafLEDsm3bW23b/l4SjSFKg+muR2G5f+R13JW0wlL17r1/f/PRJ/GNR/ZG\n",
       "Oq8btxdw9PwZZW/RfRGzbCR45st0yFtlLF/ouK05Vz2KSqGEXcU12uealsXsh4piNPhiBWTbtn8X\n",
       "wLcAFJNpDtHgCBpqrRRKuK+8IvRxLQBPrNsEoLN3bzp8/uK8wNNnT8HyWXMM6HuL8iLm3rHluLVY\n",
       "9z1GpVBqG/IOa14xJ+0AKOTyrSVcknfYOaimN1GWxO0h/xWAJwDtdBgRKcg5XZm4FYYDtAKKt3fv\n",
       "DpRHz5/pmA91Jzk5ABo+a47ddL1Fs/lds12h4tINOz+xbpO2pre8yOAGE5QFsQKyEOI0AP/LY6Ih\n",
       "pfqRD5rTTYLffGivd6QCzHaF8qN6H92FSWRhEN2w8+mZC9qa3vIigxtMUBZYjuEVso5t2w8A+K4Q\n",
       "QjtJZNv2cwC+orrvxIkTWLVqVaw2EGXVS9UPcdNpXrOOW3nsKz3Ydv/rtz5W7rzkZ9zKY1dxTdtQ\n",
       "rduL80J5u65+dRhrcmP49LL7Wn+HaX8OFu7NlVqP9x7Lj/t9VJ3HuJVv3e8VdJ/8TP6mcQtvLFwG\n",
       "AN/3lyiqq1ev4uDBg7q7v9qTgOzzvJm33noLa9eujdWGc+fOYcuWLbGOkRWDdC7AYJ1PlHPxbiyh\n",
       "6nEdeOdl40pWJrWgnz57Sht2k9jNSQ7lynPxrsO+XquGrv4V1BN1v4+6jS905zYxWsRdxTIuzrVP\n",
       "D5i+di8N+/8vWZXUuVy6dAmPPfYYAKwTQlz03p9UYZB4UZ1oAKgqQpnsInR40+M4ev5M4P9EFvRb\n",
       "A7pfW1cpC7iz/lm+lvvfpsHaWxXLu3Whu7Rm2GPpuN9H3QVHebSoDNY3bi9gZKmWNje5oCyLHZCX\n",
       "ovyvxG8KUe+ZllU8Pj2FD+avwDorlI8LWxHK+7pHNu8O7AEuL5S0x3K/dlClrPJoEY36IkZHR/HE\n",
       "uk2tBDHTLRd15yDfE2/1sqT5leb0wz2PKetYGISGlmkxCG9msupxYdaxPvvuqx2ve/L9t1EpjGmD\n",
       "il/Gb5hErUqhhGc27MC+0oO4d2w5/qN4F9drVdxuLIY6RqUw5vveHVi/3WiNddhM5uPTU6FLc7qH\n",
       "prux3pwoKaxlTQPJpOerC6LHLryJk5/6bODj5GYKYUpJ6gLKbK2qDcblfNG3PrbpfJH7ON+59d9x\n",
       "ef5OMpZpkJNDvU8r6l+7h58nyytx8lOf7Zhflo9zHyuMKPW7OTRN/YIBmQZO3A0F6k6jlWgVtD2h\n",
       "7B2q5l9Vvb8omyToaneE3XTBfZywmd3SbK0auBmFm2qYuBfDxhaaQ/wcmqZ+woBMA8d0j+Og5CfA\n",
       "PFvRG7xVPbNn333V8GjtqvXbytvD9hZv3F7A02dPRd4cwsRsrdqqkAWgI6nt+PQUri/1kE/PXAjc\n",
       "CtE70mGyXaR7iJpbL1I/4RwyDS1VScY4yqNFbdnKoLnPB8p3ae+TWzImsdGB7NHnFMX1kiq3FzQX\n",
       "b7qBg2qu/ZP56yjn71TqrRRKGLfybX/LIXNuGEH9hgGZBk4aZRBlspQuacivN1vOF/HFzbt8k6BU\n",
       "CWJxeroNOG0BuFIo4cjm3aF3cdKZrVXx9Qtvtt0WditE3Vz7XH0BQHOZ1oH127GruEZ5IcQNI6jf\n",
       "MCDTwDEtg6jaRSiKOGUWLQAHN+7A8empwDXAc7cX2v6Osg2im7dHL4eXk+opL8bo2ZsMx9edRiu4\n",
       "MnuaBgEDMg0kk12Pkhiy9ivW4aara31k825MllcaByBvgHOf57oJ/bC36rV1Pfok55hlNvrTZ09p\n",
       "L4Bkophqm0eT45++9aOOTTQAbhhB/YcBmQZS3DWnIwHbEkq6Yh1e3t6sDManZy74lrr08g65und3\n",
       "mvHsHOV3BkWMaNvtbWucPZslB82lVX5t+mD2Cn7n3T/DR3M/DX1RoJoj5oYR1G8YkCk27x60/UL3\n",
       "o1/OF422JQzb23L3ZmUwlklHcal62H7HvYVF5VCy/Cxna1Xkl8pNHt70eGJzyw78LxTm6gs4+f7b\n",
       "kbOhZY9cfg9NRkqIsoIBmWIJU+0qa0FbNwc7V18IDJJRelvuXnuYes9eK4rJ7C/sTW5yf5YA2ua0\n",
       "TStvmehF4fsPZq/gwDsvA+D8MvUPBmSKxSSTNe3lJ34XA7IHZRJslmEE5Xwxdm8rbEEPr5kb17D/\n",
       "7Km2edckEtT8PktZeSupoKxTzjf3itZdtE2MFtuWPfmpOw0cPX+Gy5yobzAgU9elufwk6GJA9lpV\n",
       "G9i7VQol/POxX8AL2/ZG6m3Ji4L9Z0/FCsZesicYtr7zuJUPfVFhkgkeR6VQwgvb9mpHD8r5Ir7x\n",
       "yF68sG0vHjacY3bQORJAlFUMyBRLljNZdT1RVQ9eN4xqoXk+K4pjeHFeRBpy9w4FJy1KkLzp1Fu7\n",
       "PElBn2XYymBhySpfutexrDsXNj9cKldKNEj4jaZYTDJZ0wjapsPCfo+TyVfuDGbZyz7wzstGQ6Fx\n",
       "h6eT8g/v39gRwIKykvNWDtdrVRx1JUl1m9+FS7V+u220Q5YrXYYRfGnzbmWAdn/PspjHQOTGgEyx\n",
       "BWWyprH8xK83Z9Lrs4BWG1WPMZmfTCoYTywV8IjqS5t349OTG5XD8t7RgkrhTsJY3Wm0DfWrlPPF\n",
       "xAqs+KkUSsqRAAfNJWqT5ZU4vOnxjupjLKNJ/YSbS1Bs3g0EVLKyObwMtEHGDRKHHADHLrzZCnTe\n",
       "zQuSGuK93WjgxmKzSpdq96lKoYR6o9EqKen2L+1trYAUNGR+fHoKF+euae/3Pj9v5XBz6TVzlmW0\n",
       "VMyUaqOOoPXak+WVOLJ5t/J7ZrrhiB/TjSq4oQVFxR4y9USvN4fXVcZ6yt4W+LhKoYSDG3f4Pkby\n",
       "9iJNh7LDuLV4Z7cn7zreidHmHscHN+5Qru89PXMBXzv/hu+wfNg5YgtobTcpz10XjKP07EesXCvw\n",
       "ynrVgP6z2lVc0/q7W9+zMMv72BOnqBiQaSCphsm/+eiT2HrPusDHeX/MD23caZxAVHcaeP78GXz9\n",
       "whuB1aYstC9XkoEuiAzKckMLoBl4VSFxtlbV9nrdw/JhLC+UArPSgTvvZZja2CNWru3Ysl71R3M/\n",
       "1X5Wd+eWBR43bh6D6UoBbmhBcTAgk1a/J8GYVmkyeZxqftIv0MzcuNaxVaCXg2Y9bVkR6/7yXcYZ\n",
       "08s9Fw5R5qq9AdykXKVpEJMXG2HKggLQznP/wV9+D0D0ylsso0n9gHPIpORNSJJDb3JXoH5gMrdt\n",
       "+rjJ8krsWXY/pho/AdAMDEHVtmZrVYwsBdvZWlX7OBmE/eZvvRadBj6a+2lrfjiq/WdPAejsrauU\n",
       "88XW+/RQZbXvuTsAfjx/PXK7vOROV6afqUqYPAbvPLDqfFUXBaaP073Oo1hufkI0cNhDJqVhGXoL\n",
       "Mwpwd25Z2/ykSbKO7PEF7cTkF7BVbtxeaH0WSSSPyc0f/NysL7QtkYoi6taOJkl2QUznl1XzwJ/M\n",
       "X8fE6J026HrYYXriqtd5qfoh55uHGAMyDa0oCTjeAG6y5eFsrYqfLcwHJjhlvdBFElWvlhdKxlW2\n",
       "JG+SXbfpLkYdB4lNgehe56ZTH7iLXjLHIWtSCjv01o90P7y6pUyv3/oYl+fnW4/9YPYKKoUSxvKF\n",
       "wL2V640G7l5W9u0JOyFrecnKVkHDx93ycMjXld+fyfJKHHjnZaP5cvcwedryuVxiUyBEKtm+JKfU\n",
       "DEISTNBwtC78eZcyyV7z5cZ8x2Nna1XkYKFSKPkmcFUXbwfOES/6rOPVDfV+MHulZ8E4zgWZO6Pb\n",
       "ryZ23srhS5t3t3qYvewZS72qLKd6nSg1xmlwMCCTVhb2ko2a6Z3ketCguXPZc3ph215t4DRZJqRT\n",
       "zhfxlL0t1SFt7xKpMFXIvOu//ea85TKnA+u3p3YB2KuLUdXr7Cs92FcXvZQsBmTS6nUxD684QdUk\n",
       "KS1sgtGaXOc+xN6LFZOlQ2HlczlsvWddrKAeV86yWhdGz777qm8w9r6vD1VWd6z/9pNU8mCcZXu9\n",
       "uhjNwkUvZQfnkCmzkih3qHr+02dPaZeyqMjdnmZq1zpuv3dsOY6eP9Ma/nZXsUrKbK3aWp6UBu/5\n",
       "+GVjyzl195IneSFVKYzhR3PXjGbKZ2tVPPvuq/i9bZ+J1Oa4y/Z6NQ/sfZ2rmOn6a1J2sYdMA8mv\n",
       "p6pbyqIig67c7UmyAJRGCh27E8ldiEwUciNGj0ub6fnIYe1PFOuPZcWwMGlrN+s1PH32FN77yUzo\n",
       "3u6wLNujwcKATKkw+YGNk1zjnZ9TkUtZdCw0hxRVP+4OgE+q8QpfFHKDM0ClqhOeBAfAt8W7xlMX\n",
       "8nvVrb2nibqJAZl6znRuOE5yzfHpKaNiG6odkqTlhVJX587zuVzkQhlp85YRddcJ78Y8upeqt+v+\n",
       "XqlwnpayjgGZei7McGKUpJck9iF2v56up/5AObgoSNDxexG84rKAtiVdlUIJR1xLk7yfi8nohNfE\n",
       "aBHjitKdYS5Y/D7zqJnS/V7PnfoLAzJlWphyh/KHM24wzls5XK9VcfT8GRyfntL21Ev5UeXzJ0aL\n",
       "HY/3/i2TwaKUvex1r9pBcyQhbzV79LO1Kp4/fwa3G4vaz0VeSJmQu1b93rbPtJ2bBeCbjz6prOzl\n",
       "vRB49t1XtceXUw9hcStF6jUGZOq5pAsveH8441IVBnli3SaMW3mj/YMdp7Nn7/57RXEscnvH8gUc\n",
       "2bwb5XwRFprB36R8ZxLk+yLdrNew/+yp1n/uHqRplvKI5+LnKXsbLLTPSZtMXfhlfh/ZvDvS1AMT\n",
       "w6jXBierhPrGoY07cfi911pzvPIHNqokNlcAmkFAt6fw6ZkL2Fd6EFu2bAk8Tj6XUwYk+XecJUxy\n",
       "1vmFbXvbbne/n2n5YPYKfufdP8PBjTtaAVBXXnMsX0Btsd6WwS0z31UBNMxOTW4W0PU19N4dm6Ju\n",
       "vEHEHjKlopcFESqFknJ+0s1CsycVZjhYN/+7othZQESKOw85V7+zy5N7mN7vNXtprr7Q2rsYaF58\n",
       "eUuKWgDuL6/U7n2s6oGenrmA67Uqri9dHEnHp6fw4rxQtiVu5rdqO0rv95XD2pQk9pApFUkWXlAV\n",
       "+Cjni7Cs5pDogfXbcfT8Ge3zZTCeLK/UFgupFMYAw1ofMzeutXrBcv5T9qCSGFK/Xqvi6xfeaFsb\n",
       "PXPjWuIFSaKSexdLBzfuwLELb7baJgOXjiyEMp4v4Pe2fUZb5GNFcaxjfbgk55+jOj491TEMLuei\n",
       "3T3ubhSvoeHFgEx9z2QIXBcI3cFYdSzp4tw1vIhrwFmBh5eGJU2Gyrux8YMDKANRFoIx0PleT2p6\n",
       "w0FkYRDdNILfEH3cNdG6tefencCIksSATJkTZU7Ob45RN0xczhfb5jvdx3rep0f9wewV/PY7L7P4\n",
       "hMaykeR+VqK8x5VCybh2dtjvmnfOWzUqwfXOFBUDMmVK1BrEfkPgup6sTL5SHUuX4CV1a6MHuSQq\n",
       "aE2tarhWd3uvLSzWO26zLAuOX1m0hJgscZJB2Nsa93fNtM553Wm0fVfiJijScGNApq5T9UJ0PZOo\n",
       "c3JJZ7qa/iAnyT1HqcvEtoBWcpO7d+YOBL/9zsup7gzlAK3121j6dyNGMFZlZOsEVVcLKhoj11h7\n",
       "NwmpFEq4XqsqL9LkbfmlfAWiqJhlTV31+q2PO7JQD7zzcqKZqUGZrlHWPR/auLOn+w9757J1HNd/\n",
       "7gC1ojjWyrpOMxhL7s8g7rK0ar1mFIwnRouBAdH0Iku+v3Ktt0lVNbmXMzOsKSoGZOqqy435jttU\n",
       "P66yFxwleAYVcIhaE/vwpseVy6C6EajHXUuDoiyNmrlxzbeOcxqSKqJhck6VQgnfeGRvYO84ymuP\n",
       "WM2pDdMNS1g4hKJiQKZMibOhhB/Tdc/utb2nZy7gyObdbVs0TowWcXjT46FrNQeR64uTqMOdNar1\n",
       "vGGpLoLK+WKzBraVNxoqTuJ9dX+PdOSe26x9TWFFmkO2bTsH4A8B/BKABQCfF0J8mGTDaDCsyY11\n",
       "9JKDMlPDVmVSzfd6A6/JumdVQtnJ99/GMxuamdjnzp1rVeoKysSOImgpT7+Rc+J+a8BNlfKjcJw7\n",
       "u3O558zPnTsXeMEWVOt6uU+inO575Hfx5J46CUpIJJKi9pB/A0BBCPErAL4A4IXkmkSD5NPL7uvo\n",
       "8Z781Gd9e8G6qkw6SfWqdUPfz58/g6fPnsLrtz5u3S4zsYfdiKV/F8bzRUyWV/quATcdabhxewGW\n",
       "Be0oh9+uTKoiH+42HNm8G8e27sEXNu0K9T3iEDYlLWpA/lUAZwBACPEegL+bWIto4KiGi3VDyFFL\n",
       "EUYtxen+Ifebq3TQnA8PShbrlizOLVUKJdw3rr/omasv+PZMS/kCDqzfbnxhM2LllDt/qRIH3Z+T\n",
       "X1KZNys77PfIZAibyJQVZW2gbdvfAvBnQogzS3//CMA6IYQyFdK27ecAfEV134kTJ7Bq1arQbaD+\n",
       "8fqtj1vD1mtyY/j0svu0j9XVJR638thXerBr7QrD3ZaXqh/iptNcd5uDhUZG0qqC1lHHNW7lUbEK\n",
       "kd6/KMdxv7fyOxT0+eVg4fNjv6j9TgHAE8vux925ZdEa76Fqz7iVx67imsReg/rb1atXcfDgQd3d\n",
       "X426Dvl/A5hw/Z3TBWMAEEI8B+A59222bT8AYGbDhg1Yu3ZtxGY0uef2+t0gnQsA/D/f/09tP1KX\n",
       "G/N4pf6Rdl7NOiuUgWR0dDT2++Jdq/zj+WjBxN2WVXPr2ua7T77/dtfmgUcsC4sBF9B5K4fyaBGV\n",
       "whguznWvQMhizsKPF+MFY6D5Xn556290lD4F0Prbm3NwuTGPb1f/V+BSqAYcvFL/CA+U71K+F+V8\n",
       "EQ8//HBi87tbsCX2LmaD9P8/z6XTpUuX5D/XCSEueu+POhL2fQD/JwDYtv0IgL+IeBwacKoejN+8\n",
       "WtJ7JUvPvvtqx7Bm0NzmA+XOfYZ1ST5yGFUOYaZFBqluBmMAuLV4O3YP3G8Kw/23KvCa1u2erVVx\n",
       "6ebPlPe5d85y85uPDtLLXcxo8ETtIb8G4O/btv39pb+fSqg9NOSS3isZ8E/qUSmPFluv6W7LuJXX\n",
       "tsXb+9Y5JknMAAAgAElEQVRVdYojqHcs9UOmtoXm/tB+Fdbkex00vx8kzKYbUUu3SknuYkbDJ1IP\n",
       "WQjhCCF+Swjxq0v//c+kG0aDYU2uc5/eoN5D0r2MsJWiRlxrXmVb8lYON516R6/p+PQU9p89Zdz7\n",
       "1kk6Y7uXVcYk03OQ+xSrEvgOvPNyRwKfatTE9Pz8RitUda9Vy5hma9W2PZ6JuiWLyZs0QFTLnoKW\n",
       "JHmHgXvJfRFwfHoKR8+fwWyt2rGX7+H3XsPXzr+RSLGJvJXDkc27Iz13RBGYKoVSV4qXqPzD+ze2\n",
       "Lp5MtjyU+xRvvWed8kKp7jRw9PyZtqCsWtZ28lOfDXytiaWRjoc12fCyVKl7iFrHu8czUTcwIFPX\n",
       "pT2vpuphWQC+tHl3R9CSa5/dvTeV2Vo1sXnautOIXGRk0Wloz+HA+u2xe8pBvd7//slf4cD67Ti2\n",
       "dY/RlodlV9UzHQfomNuN8h2SI/y6uuQn33+7dVEle+k67tKmRN3C3Z6o69KeV/POS7s3cjiwfjuO\n",
       "XXizowfcT67Xqm3LguQ5nHz/bdxXXtFWfUpVJc3PU/Y2fFu8q73fuxPXeL6gna/3BtMxn8fO1qp4\n",
       "9t1XMb90/0OV1R3foYcDduTK5+4EYV39dJP59n5M0Ep69zPqDfaQaSi4e1juXZUmyyszsTtSHA6g\n",
       "XP88W6t2lIIs5UeRCzFjbVIpze33tn1GeXTvVIVJot3Nes23QIxfpaykgmhpZDSVqZM4ohbXofQx\n",
       "INNQ6NW89MRoEWXX8GYayVV+btxeCFW8JKgHqQp8T9nbWkF5LF/QVmMLa7ZWxbELb7bddmD9doxb\n",
       "+baLAFXwj8rdy+4XQbufUXZxyJoGSpShOt3mFItOAzdCJvOMWDkc2NAsEDJ3eyHU8HDgsWFhURNM\n",
       "cwB63c+XS5e8tt6zTjufHHc3q7rTaFuGNFleiX2lB7HKXqfdkCTOfsyqpDmibuG3jQZG1KE63eYU\n",
       "z2zYgUqh1NbjNTFZXol7x5YnGowrhZJvz/bnPPWUe9EzN8mq9vILjqZtVvX2/EZAoq5h7se5Y6B7\n",
       "xXWo+9hDpkxIIgnFb6guKKlMteWjOxnNpDiF+0cvqcQwC80M36BtDL1Dy0leDOh8W7zbSviSS4ui\n",
       "foYWgJOf+mxH8l2cgiDu71RYSRSkSUs3iutQb7CHTKnLQhJK0Byzyc5OSa9VlWt2X9i2F5PllRjL\n",
       "FxI9fpI+mL1i9Bnq3sdlI6P4aO6nWFG8U0jm3rHlyl6zSW8vaNmaH1XBkH6T9lJDioY9ZEqdrmd7\n",
       "7MKbRgUgJN1csO4HKUyv/NDGnXj2nVdaOzupyKIWUYt8eLnX7IYt/5kF8jOUWex+FzXVxdsda7F/\n",
       "PH+9Yy7ftLcX1DO2ANy1rIy/uTXXdrv8vugS//plOVHaSw0pGvaQKbNkAo9pT1k3F6z6cY3SK99V\n",
       "XBO4960saqGrDmUqb+Vw4/ZCq1RnnMSkNNWdRtt7HHYof7ZWheMg8d6eA6DeWAxVRS4LIzk02BiQ\n",
       "KXV+PaewyzVMhup0mb5Br3V3bllrWDso4B7auBMTiqpUI4ZrgL2FSpLcqKLfModvLd7WTiccn57C\n",
       "i/Oio8a4yRSDLAySt3JGwZ7Liajb+uv/TBpIfgUewgqaC4677EbSlWMEmj/Sx6enWlnacm1ypVDC\n",
       "Fzbviv3acfVbIRTdSIn7s/T2WMN8p3qRAEdkggGZMkFXdznppBS/od+wr3V40+Pa/u4Hs1fw/NLG\n",
       "FA3HwQvb9rYuElT7LJM/VU9U12N9/vwZHJ+eahstCVq6ZtLT5XIi6jYGZMqEyfJKnPzUZ5Vzeqdn\n",
       "LkTeMN6ULHIRporXZHkljig2qPC6Wa9h/9lT2L/U/i9u3tWTnZgGzWytajxfK2t5y40vDm7cETj/\n",
       "HyRMjgJRFAzIlCneOeAkE2mOT08p52Ll/rzyMX7B33u/HCI3rQ4t2//Euk0MygpBc+zuXmzQPLG7\n",
       "1+ueylDN/5v2dLmciLqJy54oU7zLNeIU+3DTzR2X80W8sG2v8jEfzF7B/rOnsG7iLnxh0y68futj\n",
       "XJ6fb7v/wDsvh56DnF3aHvHY1j2+c9oWmglYwzTHqSsNquItgGFClbEepnAGlxNRNzEg01DQBj1X\n",
       "h0w3vzxz41rzh7/R+cMfN1ge2rgT+8+eUt6XG7JgHGQEVkev9MD67Th+4b9iAY2O98o9yvJDTaa6\n",
       "BeCJdZu612iiEDhkTZmWRCKN37yz6RKgML2wIKbtTzIb2kJzK8Gka1xbAL60eXdXjuu1CAcn33+7\n",
       "bbpCbi7hl3/gt2zMQfgtJom6hQGZMk21fOX60pCvKb/M6kWn0fqBN1m7GldaiUAOgAcm7sLfWn5P\n",
       "oscdzxdxeuZC4htp6DaucGdRe6nmd7tZUCUo34AoLAZkyjzvkqgkqyTduL3QSvxJcj20l2pfYEDf\n",
       "e58YLSa+PCpKpSw/lUIJBzfuSDzoXa9V8R+XNq3QUX3+Ufa8jpqcFSbZkIGbTDEgU+ZNllcqh29l\n",
       "b0n1Q+f+EQyzKYNuPbSXBeD/GFtufNxCbkQZKHTBrFq/jR/NXTM+fq+5e/pJVhED0ApyQaKuHZby\n",
       "Vg7Xa1Uc1fS4/ZhW7WK5TQqDAZn6nveHzvsjeLNe0y6m8faQ5HrooEISDoD5eq2tPObEaNF4+VMQ\n",
       "WQM6jKRe20Sl0NyVKes9Pu+ox8Ros2JafilhrtuBkuU2KQwGZOoLprWJT77/tvJH0EEzYLmDlt98\n",
       "rmkhCVkes1Io4ZkNO4yT0H7rne9iv8Eey2Ek3VP1c3HuGg6883LgEHilUEJpZLQrbYiydviZDTtw\n",
       "bOse5Zz3bK2KE9PfM3ptVu2ibmBApr6QxPzu8kKpVVkr6MfTtJCEd97S207VsOhvvfNdNJxehs+m\n",
       "pHvQQYlcE6NFHNu6B8/+0mOxXytOFneYueWbdbM9rU2rdjFwUxgMyNQ33D2ddROdCU/yh87vRzBK\n",
       "4k/Ykomynbph0bSC8ZHNu2NvC2mqUijhrmIZT589haOefY6jHEvXoz124c1Yx/YqK3bo0jGp2sVy\n",
       "mxQGAzL1DXcw/cKmXdofum78CMo5U6AZCNyJZLpymrpEtLDirvGdGC3iyObdyvfGhAWgkAtXQ2i2\n",
       "VsXFuWtGCVp+5TLlZ6d7RN1p4KXqh6HnfnWjHs9s2GF8DNOLO5bbJFMMyNS3/H7okvwRPD49hYue\n",
       "jGfZ45XzqI7ntiSSgyw0g8ThTY8r91Y2NWLl2gLGgfXbMW7ljYqilPNFPFRZjVqjHvn1g4yNFtou\n",
       "EuRWle7Pzi+H4KZTD50k5d2vWg6vd6PnGmVUhoYTS2dS3/KrK5xkzWG/dbaqodS608DR82dwf/mu\n",
       "jkAOAMtGRnFr8bbva45YOSw6jVYRlLuKZdy4bTa/6XXj9gKeXirP+VBldbOXbBVwuTHv+zwLwD9+\n",
       "8O8ErgmOa8TK4cD67a2gKqcW3ILqVs9FeG+e2bCj7TWJ0saATNQFDoDZ2jwqhVJHEAkKxsCdspmy\n",
       "1x2He+hczmOr6nJ79aKspHdu//j0VGvOWV48SAfWb8exC29qL4IOv/eaMpjrJL1RhHvjCm/biUxw\n",
       "yJr6QprVjqKW1Jy7vZDJnleYeezrtSoeUCTQJcG7B3VQEQ3Vntlus7Uqvn7hzVS+JywAQklgQKbM\n",
       "S/vHLuqSq7rT6PsCEA6Any3MRzp/dxEOL/ce1JKuiMaxC29i/9lTrf9WFMc6Hict9qDYhwoLgFAS\n",
       "OGRNPRVlWC+pPZHD8LbzwPrt+Nr5NxC2flaSu0RFkU9oC8dKYczoXEZgoQEH5dEintmwo9X7dc//\n",
       "htl/GOicp5+5YV5StNvfkyBzivl7Ih32kKln0u7pmlK189iFN0MHYz8To8VWNnGQOAVRyqPFjiVg\n",
       "YQqEyDle07rai3Awnm8PxoBZ1nsvdtvqFlXbe1WekwYHAzL1TNRhvV5XO1K1M8ntBQHgG4/sxQvb\n",
       "9morgY1YVmvIN+p5yue6N8yYrVWNN9vIW7lIG0jM1e/soCXn/o+eP4N7x5b7Lv1RVTkLw6+iWrep\n",
       "1r7r1qFzGJt0GJAp8wat2pF3TfGhjTs7gs+i47SWA02WV8aqsOXdr/hmvWb0PJm5HLVHF2VERPak\n",
       "LYS7CFJtBtLr7wkLgFBcDMjUM3F6ur38sevm0Gl5aTjXS1ca0r1XszdoBw09+222YUoeI2wd7CfW\n",
       "bTIaEVFVObt3bHmoHvm4lceqUrljedii0+jp8LC3AAjrWFNYTOqinvEWdwiT3ONdM9rNNZ/edqoS\n",
       "oyyE313J73yDjnd8eqqjDeP5IhpwUK3Xur7T00OV1R0BT7XGWjJZvyx70FJrjXSIRLi8lcNNp65M\n",
       "9Lpxe8E4oasb36c433caTuwhU08l0dPtRXKYu52HNz3eMWT+zUefVA4jl/PFtiFpueVjOV/0PV9d\n",
       "b2pFcQxPnz2lLA4yV19AITeibUveymG2VjUO1qo5W/k56aYN/AT1EHU96DCSmNvv5veJw9gUBgMy\n",
       "9VQSdX11P+TPu7Y4jOv0zAVcr1VbpStl/Wf3D6sqSL2wbW/bHslHNu/GNx99Ei9s2+t7vqpj3Tu2\n",
       "HDM3rvkGVLnRhXyOFGW506LTaAvK3jlYb3LY8emp0IlU944tD/zc426m0d4W/ZplqRtriMMksxFJ\n",
       "DMg0UJLo3ah6TCfffxu7ims6flhVPaAoFx3Hp6davUMLzYBgWjJTtrHeaLQys02C8cRoESPWndlh\n",
       "B80ep65H700O+2D2Cj6Zv96RpHZ9KVir2u/+fHQ9aO+IhBRlP+eLc9cS2+zDVL8s76PsYUAmrTTL\n",
       "VfoJSrqK27vR9ZjeWLjccXsSPX5v8Io6HzxXX8DI0lKloOA1buXxjUf2KvdmdgDkc7m2kpa6YfPZ\n",
       "WhU3bi+0vZ4D/0xuk89HXui412sf2bw7UlCWm33oAmLSyVes2kVRMSCTUpav8qOWskyb7gInbBa0\n",
       "SVBSBRl3cNtVXGP0Wu7vgZ8oFxF+gUte6LjXa+sylyW/oicOoA2Ig7asjvpX7IBs2/Ye27ZfSqIx\n",
       "lB1Zv8qXPSi/RKSodD2mcSsfecQgyQuc5YVS4NytKsi4g9vduWUAoiVexRXn8/GeVzlfbJvbj7pk\n",
       "LcnkKy53oqhiBWTbtk8AOIpo0ztEkckelHf3nyR6N7oEq580bkUOqH4XOGGCSFDWs0m5yuPTU3hx\n",
       "XiiTwbrdO3Qf3y9w+U2XuM/r4MYd2Fd6sHVM1Xpt93F1kph6kNjjpqjirkP+PoDXAOxPoC2UIbp1\n",
       "p3Gv8rux3tO7uX0SvMeUe/S6yZ2IZIlE7/m4z9VvSNe7XtVvTfJsrYqj589gLF9ozdPmlyp6ean2\n",
       "+3XPV8sLC7lUa0RxHNX3wAKQsywsKuafvUasHMbyo3Cc5ry0+/i6dbq69cmyapn3vK5ipu01D296\n",
       "HEfPn2m9h2ms/+3Gd5IGn1FAtm37NwEc8tz8OSHEK7Zt/1riraLUdaOoQdAPbVRJbTTvvVgwOaY3\n",
       "61iez+mZC4FZ0u4LHPcP+BPrNuHb4l3t87xJU3WngWMX3sThTY9jsrzS96JH1VOfqy8AaNaC9n4O\n",
       "ft8DkyIei06jGeg3qD9j93nLNdeqMB9m16bJ8koc2bw71YCY1HeShovlGFzl+lkKyPuFEP/U5zHP\n",
       "AfiK6r4TJ05g1apVsdpA3fE3jVutzOJdxTWteceoXpwXytvHrTz2lR6Mdey4Xr/1MS435ttuG7fy\n",
       "rfNW3a8zbuVx06kHPqZiFVrHXJMbw6eX3de6X/de+bEArMotw08atzpeS55H0HHdj5V03wN5e9C5\n",
       "yuP6fcYm76/7GO7He987oqy6evUqDh48qLv7qz0pnSmEeA7Ac+7bbNt+AMDMhg0bsHbt2ljHP3fu\n",
       "HLZs2RLrGFmRtXPZFfP57vOxzgpl72d0dDT1c/7W2c5AddOpY6rxExz75T3Ygi149p1XWsGnUijh\n",
       "uqYK1ujoKKxaXXmfhWZS1oriWFu5x8uNebxS/+jOZhLT143XIUsO0BGMvecRdFz3Y91034Nd6Bz5\n",
       "UAn6jFXvv5scTZAjAJfn7wRv73vXz7L2/38cPJdOly5dkv9cJ4S46L0/iWVPMs+FyFe/Z5/uKq5p\n",
       "S5LyOx/dffeX78L1WlVZe9m7mYQ7MSiprMluLBnTJVJJcT9jb1JU1lcAEEUVOyALId4WQjyZRGNo\n",
       "sGU5+9TkYuHu3LK2TFy/89Flal+c8y+F6ebOJo5zxes9D1kGNMklY7rKYBZg9Bmr3n+TGuBEg4SF\n",
       "QainslpsP+rFgt/5eO8LWtPrPYZ7KU6YHrL7sarzmCyvxL7Sg4kuGdO1rzzauU+xiq7nfl95RUd7\n",
       "oi6XIso6BmTqqSTXeyYtysWC3/mEOdegYKgKQiOaYeJ7x5Ybn0dSF0i6IKna+9mvLe5eu269tzd4\n",
       "j1t5HNu6p5XZnsXqckQmGJCJlkS9WDDtlemGZSdGg4dlVT34P/T0cKUfz18HgMAkp+PTUzh6/gyu\n",
       "16pGuzCFbV/Y93GyvLK1pttNNT/svpCQZUA5t0z9jgGZKIYwJTFVw7IPVVZj7fgKHD1/JjCgq3qz\n",
       "ukAeFIhev/Vx4r3JXk5HuC+e4i7HI8qKnix7IhpUfr0yVWGIFcWxtmIa3uVCfsVSVMUmJssrfSt7\n",
       "6ajW/IYpviFFKabiJ06FuG5VlyPqFQZkoh66qFju5OUXGFVVuEwDkfu5SehG5bU4FeK6UV3ORDfK\n",
       "wdJwYkAmiqGXvTK/AHjy/bd9A5FJ8Q6gWerya+ffwMW5OxcOD2uCjG504PnzZ2AhfHCSgc1Bsz53\n",
       "2WBu3avXNaS7VQ6WhhPnkIliCJvMZLKzky6g6wLgsQtvtoKxbqMJ057xjdsLbcEYaAaZf/2DPws1\n",
       "vxx2Xtq777JuXXOQXmfxM5GMksSATBSTKplJl3mtCuBxs5PdwavuNHDy/bcTX+pz4/ZCR5Axubgw\n",
       "DU4MbEQMyESxeXtlQZnX3gBump1sum+yKpCpnutX7tJEN8pwRvH6rY+7VgwkaElbv5eDpWxhQCZK\n",
       "mGqu1h0kvQHcdJhV1bsOquAlA8oPZ6+0BeBxK6+s1PWwJujr1kq7LybWTdzVcX+YDOkozz0+PYXL\n",
       "jfmuFAMxWdKW5XKw1H8YkIkSFLaHFrbUo7c3HVRG0h1Q6k6jVYhEFtNwH29FcUw5dFwplPCNR/Yq\n",
       "g4z7YuILm3ZFDk5RA1s3h7pNj53VcrDUf5hlTRTAnf07/u4M5us1AOosYl3ylIXOrN8oGbretch+\n",
       "S31UbXHQLLkpi2nI4+mysMfyhVBBJk6Wc68zpJOiWh9OFAUDMpEPb6C6uRSMgWYA3X/2FAD90iBp\n",
       "eaHUEWRNi4oErXNNIpDpLiQKuZHQ5S+DgpPufKIEtm4uO2OhEeo1BmQiH6bLhWTv9oGJuzr2Ojap\n",
       "Ve02d3sBTy8F+rF8QXkR8ED5Lnxx8y4A+kDmF1Cuihnj9sThDb4AEl23e2jjTjz7ziu46dQBJFsM\n",
       "JK1CIzS8OIdMlJDZWhU/W5jvmAvVzb/qMp/rTqM17+sOxm4X564FJi+FmZftRrawKikqKOHN+3yT\n",
       "+fVdxTVdm8Pl/DD1EgMykQ/TpUZupj/iqoCp2u1IxyR5KU5bdMHbJFCaVgbTCbNpx925ZV0rBpLl\n",
       "7UJp8DAgE/kIs9ZWBr0wP+Ld7oFNllfi3rHluF6r4uj5M749zUphTPlvN5NAGTYYq86dhUJoGDEg\n",
       "EwWQQXPc0qdcWACObd2D0zMXQi1j8gZv3Z7JKqbrdFUB9G8atzoe5y6ZqRsS1w05n5j+Xutvv3n3\n",
       "SqGEidFi29/sfRI1MSATBZBBc1/pQW1wLI8WA3uPJkO9qqHjI5t3dxQBibtO942Fy0aPM+2R3qwv\n",
       "BD5GXrQ8s2FH4KgAK2DRMGJAJgpBFyie2bDDN6iFmRP1DmPLCwIZmLMYmMquXq+ul/+UvQ2A2bws\n",
       "K2DRMGJAJgqhFxWldAErSoKR7gJCVuoKepw38KtKa8oLEkn1Hn3z0Sex9Z51Rm2WmOFMw4YBmSgk\n",
       "XaDI4jCr7gJCVuoKepw38B/auLNtDnhitKh8XBLBlBnONGxYGIQIwdWw3HSFOPwKSaRZ9cm0kpfp\n",
       "457ZsCPwcSwnSRQeAzINvSg1pXV0QS3Nqk+mwTHpxxFROByypqGX5JpXv2FWzokSkR/2kIl6hD1L\n",
       "IvLDHjINvSwmYxHR8GFApqHHNa9ElAUMyETg/C4RpY9zyETg/G5awiw3Ixp0DMhEXcBAoyffG8dz\n",
       "u+lyM+97+yiWd7G1RL3DIWuihIWpWz1s3O+NStByM9V7+1L1Q763NBAYkIkSpNsLmHv5NvltzRhE\n",
       "997edOp8b2kgMCATJUQXMMicLqmO7y0NA84hEyXEr/fnDTTDOsesqukt+ZUT9Xtvx608M+NpILCH\n",
       "TNRlFtC2rnmY55i9a77L+WKs5WYWgH2lB7lmnAYCAzJRQlQVvywAT9nb2m5LsnZ2P3Kv+T64cYfR\n",
       "Foum7y1RP2NAJkqIquLXNx99ElvvWZdiq7Inyj7HfG9pGDAgEyXIpOIXa2dHw2pqNOiY1EWUIJOK\n",
       "X2nujdzPWE2NBh17yEQpYG+PiLzYQyZKAXt7ROQVOiDbtr0cwJ8AmABQAPCsEOIHSTeMiIhomEQZ\n",
       "sv5XAP5cCPFrAD4H4GSSDSIiIhpGUYasfx/AwtK/RwFUk2sOERHRcPINyLZt/yaAQ56bPyeEOGfb\n",
       "9moAfwzgYLcaR0RENCwsx9FthKZn2/ZGAN8F8DtCiDcMHv8cgK+o7jtx4gRWrVoVug1ERET95OrV\n",
       "qzh4UNuH/WrogGzb9t8GcBrAPxJCTEdtmG3bDwCYeeutt7B27dqohwEAnDt3Dlu2bIl1jKwYpHMB\n",
       "But8eC7ZxHPJJp5Lp0uXLuGxxx4DgHVCiIve+6PMIR9FM7v6D2zbBoBZIQTXbxAREcUQOiALIX6j\n",
       "Gw0hIiIaZqzURURElAEMyERERBnAgExERJQBDMhEREQZwM0liDLk+PQUfjh7BUBz3+RDG3em3CIi\n",
       "6hX2kIky4vj0FD6YvQIHgAPgg9krOPzea/ho7qdpN42IeoABmSgjZM/YbbZWxcn3306hNUTUawzI\n",
       "REREGcCATJQRD1VWd9xWKZRwYP32FFpDRL3GgEyUEYc27kSlUGr9XSmUcGzrHkyWV6bYKiLqFQZk\n",
       "ogw5sH47KoUSe8ZEQ4jLnogyZLK8Ese2cq8WomHEHjIREVEGMCATERFlAAMyERFRBjAgExERZQAD\n",
       "MhERUQYwIBMREWUAlz0REYE7bVH62EMmoqHHnbYoCxiQiWjocactygIGZCIiogxgQCaiocedtigL\n",
       "GJCJaOhxpy3KAgZkIiJwpy1KH5c9ERGBO21R+thDJiIiygAGZCIiogxgQCYiIsoABmQiIqIMYEAm\n",
       "IiLKAAZkIiKiDGBAJiIiygAGZCIiogxgQCYiIsoABmQiIqIMYEAmIiLKAAZkIiKiDGBAJiIiygAG\n",
       "ZCIiogxgQCYiIsoABmQiIqIMYEAmIiLKgHzYJ9i2PQ7gFIAKgBqAfyGE+HHSDSMiIhomUXrInwfw\n",
       "P4QQ2wH8CYDfTbZJREREwyd0D1kIccK2bRnI7wfws2SbRERENHx8A7Jt278J4JDn5s8JIc7Ztv0W\n",
       "gA0AHo/42iMAcOXKlYhPv+Pq1au4dOlS7ONkwSCdCzBY58NzySaeSzbxXDq54t2I6n7LcZzIB7dt\n",
       "2wbwuhDiFwIe9xyAr0R+ISIiosH21dAB2bbtLwK4JIT4Y9u21wL4r0KIh8K+sm3bRQC/DOATAIth\n",
       "n+8xA2BdzGNkxSCdCzBY58NzySaeSzbxXDqNALgXzTysBe+dUQLyPQD+CMCypYMfFkK8m0BDI7Nt\n",
       "2xFCWGm2ISmDdC7AYJ0PzyWbeC7ZxHMJL0pS108A/IMutIWIiGhosTAIERFRBjAgExERZcCgBOSv\n",
       "pt2ABA3SuQCDdT48l2ziuWQTzyWkWMueiIiIKBmD0kMmIiLqawzIREREGcCATERElAEMyERERBnA\n",
       "gExERJQBDMhEREQZELp0ZpbZtv0QgB8AuEcIUUu7PVHYtj0O4BSACoAagH8hhPhxuq2Kxrbt5QD+\n",
       "BMAEgAKAZ4UQP0i3VfHZtr0HwGeEEPvSbktYS3uZ/yGAXwKwAODzQogP021VPLZtbwXwdSHEjrTb\n",
       "EpVt26MAvo3mHvNFAP9WCPFf0m1VNLZtjwD4FoBfBOAAeFoI8X66rYpnaQ+HcwAeE0L8z269zsD0\n",
       "kG3b/jkALwC4lXZbYvo8mjuBbEczmP1uyu2J418B+HMhxK8B+ByAk6m2JgG2bZ8AcBRAvxbN/w0A\n",
       "BSHErwD4Apr/z/Qt27Z/F80f/2LabYlpH4CrQoi/B2A3gH+Xcnvi+HUADSHEpwD8GwDPp9yeWJYu\n",
       "lv4DgJvdfq2BCMi2bVtovmFfBFBNuTmxCCHkDz7QvFr+WYrNiev3Aby49O9R9Plns+T7AH4L/RuQ\n",
       "fxXAGQAQQrwH4O+m25zY/grAE+jfz0P6UwBfXvp3DkA9xbbEIoT4zwD2L/35APr7NwwA/l8A/x7N\n",
       "rYK7qu+GrG3b/k0Ahzw3/wjAy0KIv7BtG+iT/zk15/I5IcQ527bfArABwOO9b1l4AeeyGsAfAzjY\n",
       "+5ZF43M+r9i2/WspNCkpPwfgf7v+XrRtOyeEaKTVoDiEEKdt234g7XbEJYS4CQC2bU+gGZy/lG6L\n",
       "4hFCLNq2/R0AewB8JuXmRGbb9ufQHLl407btL6LLsWUgSmfatv2/AFxa+vMRAO8tDZP2Nbt5dfG6\n",
       "EOIX0m5LVLZtbwTwXQC/I4R4I+32JGEpIO8XQvzTtNsSlm3bLwD4gRDiT5f+/lgIcV/KzYplKSB/\n",
       "VwixLe22xGHb9n0ATgM4KYT4TsrNSYRt2z8P4D0ADwsh+m6EzLbtt9GcB3cAbAIgAPzfQoi/7sbr\n",
       "9V0PWUUI8bfkv23bnkGf9CpVlq7CLgkh/hjNOYu+Hbqybftvo3m1/4+EENNpt4cANIfc/y8Af2rb\n",
       "9qMKAF8AAADVSURBVCMA/iLl9hBagetNAL8thPhe2u2Jw7btfwZgrRDia2hOUzWW/us7S7k8AADb\n",
       "tr+H5oV4V4IxMCAB2aPfu/z/H4A/sm37XwIYAfBUyu2J4yia2dV/sDSVMCuE2JNukxIhr5j70WsA\n",
       "/r5t299f+rufv19u/fp5SEcALAfwZdu25VzyPxBC9GOS6qsAvrPUuxwFcFAIsZBym/rCQAxZExER\n",
       "9buByLImIiLqdwzIREREGcCATERElAEMyERERBnAgExERJQBDMhEREQZwIBMRESUAf8/3oLW+aIM\n",
       "7wgAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f9ea590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mu = 0\n",
    "sigmasq = 1\n",
    "sd = np.sqrt(sigmasq)\n",
    "# Generate complex gaussian r.v. samples\n",
    "x = np.random.normal(loc = mu, scale = sd/np.sqrt(2), size = 1000)\n",
    "y = np.random.normal(loc = mu, scale = sd/np.sqrt(2), size = 1000)\n",
    "z = x + 1j*y\n",
    "h = plt.plot(np.real(z), np.imag(z), 'o')\n",
    "plt.axis('equal')\n",
    "plt.grid(True)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Express $z = |z|e^{j\\phi}$. The magnitude $|z|$ is Rayleigh distributed while the phase $\\phi = \\operatorname{arg}(z)$ is uniform over the interval $[-\\pi, \\pi)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10f2e5290>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": [
       "iVBORw0KGgoAAAANSUhEUgAAAf4AAAFbCAYAAAA0vux4AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n",
       "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3X+cHWV96PFPEtkgdAlqgWChJIj53tqWi+gVpBaCIl6q\n",
       "iFr7ai3FCmoFqReqLZWoZNMrSqVwhaq8bBRTvVxsQbAiYrmFmChWqZZbGsEvQncp1gQiJpBAyM+9\n",
       "f8xsc1g2u+ecnfNjz/m8Xy9ee+aZmed8Zxj4nmfmmeeZNTo6iiRJ6g+zOx2AJElqHxO/JEl9xMQv\n",
       "SVIfMfFLktRHeibxR8RQp2OYKTxX9fE81c9zVR/PU308T/Vr5lz1TOIHlnY6gBnEc1Ufz1P9PFf1\n",
       "8TzVx/NUv4bPVUsTf0QcExErx5X9bkR8u2b5nRHxTxHxjxHx2lbGI0lSv2tZ4o+IC4DlwNyashcD\n",
       "Z9UszwfeAxwHvAb4aEQMtComSZL6XStb/PcDbwJmAUTE84CLgfPHyoCXAXdk5vbMfLzc58gWxiRJ\n",
       "Ul97VqsqzswbImIBQETMBj4LvBd4qmaz/YDHapY3AfMa/a6ImFv+fQGws8mQ+8rYvxtNzvNUP89V\n",
       "fTxP9fE81WUOFDkwM7fWu1PLEv84LwGOAK4C9gZeFBGXAyuBwZrtBoENk1VU9mDcU2eG+6cdaf8Y\n",
       "7nQAM4TnqX6eq/p4nurjearfUxExvmxZZg5NtPGsVo7VX/5iuzYzX15Tdhjwxcx8efmM/1bgv1H8\n",
       "IPgO8F8zc1uD3/MC4P5rrrmG+fPnVxa/JEndat26dZx++ukAR2TmA/Xu144W//hfFrPGyjJzXURc\n",
       "CXyTor/BkkaTfmknwPz58znkkEOmE6skSTNNQ4+4W5r4M3OEosf+Hssy8zPAZ1oZhyRJKvTSAD6S\n",
       "JGkKJn5JkvqIiV+SpD5i4pckqY+Y+CVJ6iMmfkmS+oiJX5KkPmLilySpj5j4JUnqIyZ+SZL6SLtm\n",
       "59MMtm3bNkZGRiqvd8GCBQwMDFReryRpz0z8mtLIyAjn33Q1gwcfUFmdm9au5+OnnsWiRYsqq1OS\n",
       "NLWWJv6IOAa4JDNPjIijgCspZhHaCrw1Mx+JiHcCfwDsAD6cmTe3MiY1Z/DgA9j/0IM7HYYkaZpa\n",
       "9ow/Ii4AlgNzy6KPA3+YmScCNwB/GhEHAe+hmK3vNcBHI8J7v5IktUgrO/fdD7wJmFUu/05m3l1+\n",
       "3gvYArwMuCMzt2fm4+U+R7YwJkmS+lrLEn9m3kBx+35seR1ARBwHnAv8L2A/4LGa3TYB81oVkyRJ\n",
       "/a6tnfsi4reBJcBvZOajEfE4MFizySCwYYo6hoClLQtSkqSZZTgixpcty8yhiTZuW+KPiN+j6MS3\n",
       "ODPHkvudwMURMRfYG/glYM1k9ZQHMjSu7gXAcLURS5I0IyzMzJF6N25H4h+NiNnAFcCDwA3lL5Nv\n",
       "ZOayiLgS+CbFY4clmbmtDTGpw3bu2MHwcPW/1RwbQJIm19LEX/4COa5cfN4etvkM8JlWxqHu8+T6\n",
       "DVyxfiWDj056g6chjg0gSVNzAB91jGMDSFL7OVa/JEl9xBZ/j2nFuPqteBYvSeoME3+PacW4+uvu\n",
       "TuYf+YxXRSRJM5CJvwdV/ex809r1ldUlSeosn/FLktRHTPySJPURE78kSX3ExC9JUh/ZY+e+iFhZ\n",
       "x/6jmfnKCuORmuYwwJI0tcl69Z9Z/h0FZo1bN1GZ1FEOAyxJU9tj4h+b6Sci3gt8onbynIj488z8\n",
       "09aHJzXGYYAlaXL1vMf/YeC3I+I3M/PHZdnJwJSJPyKOAS7JzBMj4ghgBbCLYurdczNzNCLeSTFd\n",
       "7w7gw5l5cxPHMSM5yp4kqd3qSfwJ/DmwKiLOzMzVFLf6JxURFwC/B2wuiy6nmHZ3dURcBZwWEd8B\n",
       "3gO8BHg28K2I+L/9MjWvo+xJktqtrpH7MvOGiLgfuD4irgC217Hb/cCbgC+Uy0eXPxoAbqG4a7AT\n",
       "uCMztwPby+84EvheA8cwoznKniSpnep+nS8z7waOA34LOKqO7W+guH0/prYz4CZgHrAf8NgE5ZIk\n",
       "qQXqafG/YexDZv40Il4NvLmJ79pV83k/YCPwODBYUz4IbJiskogYApY28f2SJPWi4YhnPOJdlplD\n",
       "E2082Xv8v1/zuXbVKDCrXD+amZ+vM7C7IuKEzFwFnALcBtwJXBwRc4G9gV+i6Pi3R+WBDI2LdQFg\n",
       "rzZJUj9aOPYmXj0ma/EvpI5OfHUYq+N9wPKIGADuAa4ve/VfCXyT4rHDkn7p2CdJUidM9h7/0HQr\n",
       "L3+BHFd+/hGweIJtPgN8ZrrfJUmSpuZY/ZIk9RETvyRJfWSPiT8ivlD+fXv7wpEkSa00Wee+Xy+H\n",
       "0/1gRGzn6e/hN9KbX5IkdYnJEv+7Kd7X/zngxAnWm/glSZphJuvV/zXgaxHxjrLnvSRJmuHqGbnv\n",
       "KxHxt8Cryu1vB87OzIdbGpkkSapcPb36P00xwt7hwGHAPwKfbWVQkiSpNepp8R+emW+sWf5YRLy1\n",
       "VQFJkqTWqafFvysifnFsISIOAxxWV5KkGaieFv+HgG9HxJ3l8rHAH7QuJKm3bdu2jZGRkcrrXbBg\n",
       "AQMDA5XXK6m3TJn4M/OrEXE08DKKd/nPsWOf1LyRkRHOv+lqBg8+oLI6N61dz8dPPYtFixZVVqek\n",
       "3lRPi5/MfAT46nS/LCJmU0zIswjYBbwT2AmsKJfXAOdmZhWzAkpda/DgA9j/0IM7HYakPlRX4q/Q\n",
       "ycC+mfmKiDgJ+EgZw5LMXB0RVwGnAV9uc1zShHbu2MHw8HCldVZdnyQ1ot2JfwswLyJmAfMoOgke\n",
       "k5mry/W3UPw4MPGrKzy5fgNXrF/J4KNrKqtz3d3J/COjsvokqRFTJv6IOBD4HeA5NcWjmflnTXzf\n",
       "HcDewA+B5wGnAsfXrN9M8YNA6hpV35bftHZ9ZXVJUqPqafF/DbgbeLCmbNYetp3KBcAdmfmBiDgE\n",
       "WAnsVbN+ENg4WQURMQQsbfL7JUnqNcMRz7iLuCwzhybauJ7EP5qZZ003qtK+wOPl5w3l998VESdk\n",
       "5irgFOC2ySooD2SotiwiFgA+OJUk9aOFmTlS78b1JP4vl9Pz3gbsGCvMzH9vPDYuBT4XEd+kaOlf\n",
       "CHwfWB4RA8A9wPVN1CtJkupQT+KfB7wf+Om48oWNfllmbgTeOMGqxY3WJUmSGldP4n8zcGBmbml1\n",
       "MJIkqbXqGav/AeC5rQ5EkiS1Xr3v8d8TEWvYPTnPaGa+skUxSZKkFqkn8V88QZlD6kqSNANNeas/\n",
       "M79B8QreTorx9GcDL2htWJIkqRXqGbnv88DLKUbauwc4CrgJuLq1oUmSpKrV07nveOCXgeuAdwHH\n",
       "1LmfJEnqMvUk8J9k5jbgXuDIzPwBcFhrw5IkSa1QT+e+/4iICylG7vtYOR7w/i2Nqgtt27aNkZGR\n",
       "Sut0elZJUrvVk/jPAl6bmXdGxJcoZuo7p7VhdZ+RkRHOv+lqBg8+oLI6nZ5VktRu9ST+L2XmyQCZ\n",
       "+ZfAX7Y2pO7l9KySpJmunmf8z46IX2x5JJIkqeXqafEfAIxExCPA2Hj9o5l5eDNfWPYXOJVidr5P\n",
       "AHcAKyjGCFgDnJuZDhAkSVIL1NPifw1wOHAsxSx6J5b/NCwiFgMvz8zjyroOBy4DlmTm8cAs4LRm\n",
       "6pYkSVOrp8W/mN1D9M6iaJlviYjBzFzT4PedDPxrRHwZ2A/4E+Dtmbm6XH9Luc2XG6xX6ms7d+xo\n",
       "yVsiCxYsYGBgoPJ6JXVOPYn/9cCLKZLxLOC1wE+AfSPi2sy8vIHvOwA4FHgdRWv/prLOMZuBeQ3U\n",
       "Jwl4cv0Grli/ksFHG/0tvmeb1q7n46eexaJFiyqrU1Ln1ZP4DwaOzsyNABGxFPgqcBzwfaCRxP9T\n",
       "4N7M3AHcFxFPAb9Qs34Q2DhZBRExBCxt4DulvlD1WyeSZozhcoydWssyc2iijetJ/D9P0RIfswV4\n",
       "bmZuj4hdDQb3LeA84PKIeD6wD3BbRJyQmauAUygGCtqj8kCGassiYgHgaDiSpH60MDNH6t24rvf4\n",
       "gdsj4m+AOcBvAjdGxFuBtY1Elpk3R8TxEXEnRcfCdwMjwPKIGKCYBOj6RuqUJEn1mzLxZ+aFEXEq\n",
       "cBLF1LyXZOYtEXEs8LuNfmFm/ukExYsbrUeSJDWunhY/FAl/R7n9AEBmfqdVQUmSpNaY8j3+iLiA\n",
       "ojPdgxTP0T8QER9odWCSJKl69bT4zwBelplbACLir4B/Bi5uZWCSJKl69YzcNwt4qmb5KWB7a8KR\n",
       "JEmtVE+L/3bg+ohYQfEj4PfLMkmSNMPUk/jPA84B3kpxh+B24NOtDEqSJLVGPYn//Zn5UeBTYwUR\n",
       "8RFgScuikiRJLbHHxB8RlwAHAa+PiBeye0z9Z1HM1GfilyRphpmsxX8D8CLgVcAqdif+7cCftTgu\n",
       "SZLUAntM/Jl5J3BnRNwIPJWZW8uWfwAPtCtASZJUnXpe5zuPYiz9wyha/udj5z5JkmakehL/acA7\n",
       "gbcA12TmScDRLY1KkiS1RD29+ueUt/lfB3woIuZQTKfbtIg4EPg+Rf+BXcCK8u8a4NzMHJ1O/ZIk\n",
       "aWL1tPj/ISLWAHMpbvV/A7ip2S+MiL0oHhU8QdFh8HJgSWYeXy6f1mzdkiRpclMm/sz8Y+A3gGMz\n",
       "cxdFi/yCaXznpcBVwNpy+ejMXF1+voVi+l9JktQCk73Hvywzl0bE52rKxv6OZuZZjX5ZRLwNWJ+Z\n",
       "t0bEhRQt/Fk1m2wG5jVar6Tq7dyxg+Hh4crrXbBgAQMDA5XXK6k+kz3j/175d9UE65p9Bn8mMBoR\n",
       "JwFHAX8NHFCzfhDYOFkFETFEMU2wpBZ6cv0Grli/ksFH11RW56a16/n4qWexaNGiyuqUxPBYw7zG\n",
       "sswcmmjjyd7jv6n8u6KqyDLzhLHPEbESOBu4NCJOyMxVwCnAbVPUMQQM1ZZFxAKg+qaJ1OcGDz6A\n",
       "/Q89uNNhSJrcwswcqXfjyW7176pZHOXpt+RHM3NO47E9wyjwPopxAgaAe4DrK6hXkiRNYLIW/392\n",
       "/IuIuzLzxVV+cWaeWLO4uMq6JUnSxOp5nU+SJPUIE78kSX3ExC9JUh+ZrHNfbS/5549bHs3Mw1sX\n",
       "lqRe5NgAUudN9h7/iZOsk6SGOTaA1HmT9eofaWMckvqEYwNIneUzfkmS+oiJX5KkPmLilySpj5j4\n",
       "JUnqIyZ+SZL6iIlfkqQ+Mtl7/JWLiL2Aq4HDgLnAh4F7gRXALmANcG5mjk7ne6796o2sWffQ9IId\n",
       "52fDD8Evz6+0TkmS2q2tiR84HVifmWdExHOAfwHuApZk5uqIuAo4DfjydL7kJxse5afx89OPtsaG\n",
       "fx9+2rzEkiTNRO2+1X8dcFHNd28Hjs7M1WXZLcBJbY5JkqS+0dYWf2Y+ARARgxQ/Aj4I/EXNJpuB\n",
       "eZPVERFDwNIWhShJ0kwzHBHjy5Zl5tBEG7f7Vj8RcShwA/DJzLw2Ij5Ws3oQ2DjZ/uWBDI2rcwFQ\n",
       "/cwfkiR1v4WNDLPf1lv9EXEQcCtwQWauKIvviogTys+nAKsn2leSJE1fu1v8Syhu5V8UEWPP+s8D\n",
       "royIAeAe4Po2xyRpBmvVVL/gdL/qTe1+xn8eRaIfb3E745DUO1ox1S843a96V9uf8UtS1ZzqV6qf\n",
       "iV+SJtCqRwg+PlCnmfglaQKteITg4wN1AxO/JO2BjxDUi5ykR5KkPmLilySpj3irX5LaxA6D6gYm\n",
       "fklqEzsMqhuY+CWpjewwqE4z8UvSDObjAzXKxC9JM5iPD9QoE78kzXA+PlAjuiLxR8Rs4FPAkcBW\n",
       "4B2Z+UBno5Kk/uTjg97WFYkfeAMwkJnHRcQxwGVlmSSpzVrx+OCxh9bxR0edxMKFCyurE6r/MbFt\n",
       "2zZGRkYqq29MN/3o6ZbE/2vA1wEy87sR8dIG958DsG7dOgA2bXyMLT/dVmmA2554kifuG2Hrxk2V\n",
       "1fnYyE/YvukJ67RO6+yyOltV70yqc5/nzWPrs/eurM5NP36YZQ9+nmc/d15ldW752WP84Stey6GH\n",
       "HlpZnQ899BCf+NbNlcf55285u/IfPWM5jzIH1mvW6OhopYE0IyKWA1/KzK+Xyw8CCzNz1wTbDgFL\n",
       "2xuhJEkzyrLMHJpoRbe0+B8HBmuWZ0+U9AHKAxmqLYuIucBTwBHAztaE2FOGgWp/evYmz1P9PFf1\n",
       "8TzVx/NUnznA/cDembm13p26JfHfAZwKXBcRxwJ3N7JzZm6NCOwQWJ/yXI10Oo5u53mqn+eqPp6n\n",
       "+nie6leeq7qTPnRP4r8ReHVE3FEun9nJYCRJ6lVdkfgzcxQ4p9NxSJLU65yWV5KkPtJLiX9ZpwOY\n",
       "QTxX9fE81c9zVR/PU308T/Vr+Fx1xet8kiSpPXqpxS9JkqZg4pckqY+Y+CVJ6iMmfkmS+oiJX5Kk\n",
       "PmLilySpj3TFyH2NiIjZwKeAI4GtwDtqx+iPiFOBDwE7gKsz8zMdCbTD6jhPfwS8HVhfFr0rM+9r\n",
       "e6BdIiKOAS7JzBPHlXs91ZjkPHk9lSJiL+Bq4DBgLvDhzLypZr3XVKmOc+V1BUTEHGA5sAgYBc7O\n",
       "zB/UrG/omppxiR94AzCQmceV/xO6rCwbu4guB14KPAncERFfycxHOhZt5+zxPJWOBs7IzLs6El0X\n",
       "iYgLgN8DNo8r93qqsafzVPJ62u10YH1mnhERzwH+H3ATeE1NYI/nquR1VXgdsCszXxERJwAXM428\n",
       "NxNv9f8a8HWAzPwuxcGO+SXg/sx8LDO3A98Cjm9/iF1hsvME8BJgSUR8MyLe3+7gusz9wJuAWePK\n",
       "vZ6ebk/nCbyeal0HXFR+nk3RChvjNfV0k50r8LoCIDP/DnhXubgA2FCzuuFraiYm/v2Ax2uWd5a3\n",
       "tcfWPVazbhMwr12BdZnJzhPAtRQX0iuBV0TEa9sZXDfJzBt45v9wwOvpaSY5T+D19J8y84nM3BwR\n",
       "gxSJ7QM1q72makxxrsDr6j9l5s6IWAFcCfyfmlUNX1MzMfE/DgzWLM/OzF3l58fGrRvk6b+M+slk\n",
       "5wngisz8WfkL8WbgxW2Nbmbweqqf11ONiDgUuB34fGZ+sWaV19Q4k5wr8Lp6msx8G8Vz/uUR8eyy\n",
       "uOFraiY+478DOBW4LiKOBe6uWfdD4IXls6InKG53XNr+ELvCHs9TRMwD7o6IF1E8E3ol8NmORNnd\n",
       "vJ7q4PX0dBFxEHAr8O7MXDlutddUjcnOldfVbhFxBnBIZn4U2ALsoujkB01cUzMx8d8IvDoi7iiX\n",
       "z4yItwA/l5nLI+K9wN9T3M34bGau7VSgHTbVeXo/sJKix/8/ZObXOxVoFxkF8Hqa0kTnyetptyUU\n",
       "t1ovioix59fLgX29pp5hqnPldVW4HlgREauAvYDzgDdGRFP/n3J2PkmS+shMfMY/oYgY6nQMreTx\n",
       "zWwe38zVy8cGHt9M18zx9UziB5Z2OoAW8/hmNo9v5urlYwOPb6Zr+Ph6KfFLkqQptL1zX0RcSNHb\n",
       "fC/gExS9z1dQ9FJcA5ybmXY8kCSpBdra4o+IxcDLM/M4YDFwOMVQsksy83iKEcFOa2dMkiT1k7b2\n",
       "6o+Ij1C8CvTLFKMN/Qnwd5l5SLn+9cDJmfmHDdY7F3gKOALYWWnQ3WMYWNjpIFrI45vZevn4evnY\n",
       "wOObyeZQDKW9d2ZurXendt/qPwA4lGLCgcMpJmOoHfd7M1MMNVj2YNxTZ4b7px9iVxvudAAt5vHN\n",
       "bL18fL18bODxzXRPRcT4smWZOTTRxu1O/D8F7s3MHcB9EfEU8As16weBjZNVUB7IUG1ZRLwAuP+a\n",
       "a65h/vz5lQYsjRkeHuayf72N/Q4+sOk61v7Lvex74POmVQfA42sf4X2/+ioWLuzVhoykqaxbt47T\n",
       "Tz8d4Ijaaden0u7E/y2KEYcuj4jnA/sAt0XECZm5CjgFuK2JencCzJ8/n0MOOaSyYKVaTz75JHMf\n",
       "GuTZP79/03XsNbgvc/efXh0AW7ds4aCDDvJ615S2bdvGyMhIJXUtWLCAgYGBSupSpRp6xN3WxJ+Z\n",
       "N0fE8RFxJ0XHwncDIxQTDgwA91AMTShJqsDIyAjn33Q1gwcfMK16Nq1dz8dPPYtFixZVFJk6pe2v\n",
       "82Xmn05QvLjdcUhSvxg8+AD2P/TgToehLuEAPpIk9RETvyRJfcTEL0lSHzHxS5LUR9reuU/9wVeI\n",
       "1AlVXXdec+plJn61hK8QqROquO685tTrTPxd4kf/9gAP/vihadfz0iOPYv/9pzc4TFV8hUid4HXX\n",
       "+7yzMz0m/i7xNytv4UcH7TWtOrY9sYVt27fzG696dUVRSVL38c7O9Jj4u8ScZ+3F3vMGp1XH7Dlz\n",
       "KopGkrqbd3aaZ69+SZL6iIlfkqQ+YuKXJKmPmPglSeojJn5JkvqIiV+SpD7S9tf5IuKfgcfKxX8D\n",
       "PgqsAHYBa4BzM3O03XH1gp3bd/Afj/yY++67b9p19evAFpLU69qa+CNib4DMPLGm7CvAksxcHRFX\n",
       "AacBX25nXL1i88M/5cbHN/MP3/vZtOrp54EtJKnXtbvF/1+BfSLi78vv/gBwdGauLtffApyMib9p\n",
       "DmohSZpMu5/xPwFcmpmvAc4Grhm3fjMwr80xSZLUN9rd4r8PuB8gM38UEY8CL65ZPwhsnKyCiBgC\n",
       "lrYqQEmSZpjhiBhftiwzhybauOHEHxEDwO8CrwdeSNEp736K2/NfzMztk+x+JnAkcG5EPJ8i0d8a\n",
       "ESdk5irgFOC2yb6/PJChcTEtAIYbPRZJ0+dMaROr4rxs27YNYFrnZXjY/zX2gYWZOVLvxg0l/oh4\n",
       "LfBB4FvA54B/B7YDC4ETgf8REf8zM7+yhyo+C3wuIsae6Z8JPAosL39Q3ANc30hMkjrLmdImVsV5\n",
       "WXd3su8Bz512HfOPfEZrUH2s0Rb/C4HjJ2jV3wPcXCbvP9zTzpm5AzhjglWLG4xDqltVLVJbTntm\n",
       "p9KJTfe8bFq7vpI6pFoNJf7M/DhARLwkM79fuy4ifiszrwMurzA+adqqaHmBLSdJvaHZzn3/FBGf\n",
       "BM7PzJ1l2RLgumrCkqpVRYvUlpOkXtDs63w/AHYCt0fE9JpRkiSpbZpN/Dsy83zgr4F/jIiXADuq\n",
       "C0uSJLXCtN7jz8yrI+KHFD3x960mJEntYKfHie3csaOSY/K8PFMVrydC753bdms28V889iEzvx0R\n",
       "xwMfqiYkSe1gp8eJPbl+A1esX8ngo2umVY/n5ZmqeD1xrJ5eOrft1uh7/O8BPpWZT3vXPjMfAv4g\n",
       "Ip4FvDszr6wwRkktYqfHiXleJtYNryeO1aPmNdrifxBYHRGrgNXAjyme7S+gGMDnldTcDZAkSd2l\n",
       "oc595Yh8r6QYovds4IvA3wLvAhL49cx0Zj1JkrpUw8/4M3MrcHX5jyRJmkGa6twXESuBUWBWWbQL\n",
       "2ALcC3wkMzdUE54kSapSs7367wW2UbT6Z1HM1ncIsJZiIp43VRKdpAlV8WqVr0RJ/anZxH9sZh5d\n",
       "s/wvEfG9zDw9IiaahEdShap6tcpXoqT+02zi3ysifiUz1wBExK8AsyNiH6B3JtTuU93UmuymWLpN\n",
       "L83a5qA5Uvs0m/jfA3wtIh6heDPgOcDvAUuBz1cUmzqkm1qT3RSLWsdBc6T2aTbx7wu8APhVisl6\n",
       "7s3MbRHx7cwcnWrniDgQ+D7wKoqOgSvKv2uAc+upQ63VTa3JbopFrePALlJ7NJv4L83Mm4F/ri2s\n",
       "M+nvBXwaeIKiY+DlwJLMXB0RVwGnAY4FIElSCzSb+B+IiKuB7wJPlWWjmVnPbf5LgauAC8vlozNz\n",
       "dfn5FuBkTPySJLVEs9PyPlrueyywmGK43hOn2iki3gasz8xby6JZ7B4LAGAzMK/JmCRJ0hSaavFn\n",
       "5tvGl5U9+qdyJjAaEScBRwF/DdRO0zQIbJysgogYouhEKEmSYDjiGZ1al2Xm0EQbNzty35uBiyg6\n",
       "+c0G5gBzgYMm2y8zT6ipYyXFeP+XRsQJmbkKOAW4bYo6hoChcfEsAHyPR5LUjxZm5ki9Gzf7jP9j\n",
       "wDuA91LMxvcaitv0jRoF3gcsj4gB4B7g+sl3kSRJzWo28W/IzNsj4jhgXmYORcQdwF/UW0Fm1vYJ\n",
       "WNxkHJIkqQHNdu57MiIWAT8EFkfElLf5JUlS5zWb+D9IcYv/JopBeB7GV/AkSep6zfbqXwWsKhf/\n",
       "W0Q8NzN/Vl1YkiSpFZpt8T+NSV+SpJmhksQvSZJmhoYSf0R8ofz79taEI0mSWqnRZ/y/HhHvBD4Y\n",
       "Edt5+nC79Y7VL0mSOqTRxP9u4M3AzzHx2PwmfkmSulhDiT8zvwZ8LSLekZmfaVFMkiSpRZodue8r\n",
       "EfG3FO/wPwu4HTg7Mx+uLDJJklS5Znv1fxq4EzgcOAz4R+CzVQUlSZJao9kW/+GZ+caa5Y9FxFur\n",
       "CEiSJLVOsy3+XRHxi2MLEXEYsK2akCRJUqs02+L/EPDtiLizXD4W+INqQpIkSa3S7Fj9X42Io4GX\n",
       "UbzLf44d+yRJ6n7NtvjJzEeArzayT0TMAZYDi4BR4GxgK7AC2AWsAc7NzNFm45IkSXvW7rH6Xwfs\n",
       "ysxXUEzt+xHgMmBJZh5PcffgtDbHJElS32hr4s/MvwPeVS4uADYAL8nM1WXZLcBJ7YxJkqR+0tSt\n",
       "/og4EPgd4Dk1xaOZ+WdT7ZuZOyNiBfAG4LeAV9es3gzMayYmSZI0tWaf8X8NuBt4sKZs1h62fYbM\n",
       "fFtEHEQxCNDeNasGgY2T7RsRQ8DSuiOVJKm3DUfE+LJlmTk00cbNJv7RzDyr0Z0i4gzgkMz8KLAF\n",
       "2Al8LyJOyMxVwCnAbZPVUR7I0Lh6FwDDjcYjSVIPWJiZI/Vu3Gzi/3I5Pe9twI6xwsz89yn2ux5Y\n",
       "ERGrgL2A84AfAssjYgC4p9xGkiS1QLOJfx7wfuCn48oXTrZTZm4BfnuCVYubjEOSJDWg2cT/ZuDA\n",
       "MpFLkqQZotnX+R4AnltlIJIkqfWaHrkPuCci1rB7cp7RzHxlBTFJkqQWaTbxXzxBmcPsSpLU5Zq6\n",
       "1Z+Z3wAep3gdb1dZzwuqC0uSJLVCsyP3fR54OfA8ilfwjgJuAq6uLjRJklS1Zjv3HQ/8MnAdxdj7\n",
       "x0yjLkkdjnixAAAJ00lEQVSS1CbNJuufZOY24F7gyMz8AXBYdWFJkqRWaLZz339ExIUUI/d9rBwj\n",
       "eP/KopIkSS3RbIv/LGA4M+8EvkQxU985lUUlSZJaotkW/5cy82SAzPxL4C+rC0mSJLVKsy3+Z0fE\n",
       "L1YaiSRJarlmW/wHACMR8QjF9LpQjNx3eDVhSZKkVmg28b8GmFV+Hi0/O3KfJEldrtnEv5jdiX4W\n",
       "xeh9WyJiMDPXVBGYJEmqXrOJ//XAi4EvUyT+1wI/AfaNiGsz8/KK4pMkSRVqNvEfDBydmRsBImIp\n",
       "8FXgOOD7wISJPyL2ohjW9zBgLvBhikGAVlDcNVgDnJuZPjaQJKkFmu3V//PA5prlLcBzM3M7RQLf\n",
       "k9OB9Zl5PPDfgU8ClwFLyrJZwGlNxiRJkqbQ9Hv8wO0R8TfAHOA3gRsj4q3A2kn2uw64vvw8G9hO\n",
       "cedgdVl2C3AyxSMESZJUsaYSf2ZeGBGnAidRTM17SWbeEhHHAr87yX5PAETEIMWPgA8Cf1GzyWZg\n",
       "3mTfHRFDwNJm4pYkqQcNl0Pn11qWmUMTbdxsix+KhL+jrGMAIDO/M9VOEXEocAPwycy8NiI+VrN6\n",
       "ENg42f7lgQyNq3MBMFx/6JIk9YyFmTlS78ZNPeOPiAsoWt0PUiTcD0TEB+rY7yDgVuCCzFxRFt8V\n",
       "ESeUn08BVk+0ryRJmr5mW/xnAC/LzC0AEfFXwD8DF0+x3xKKW/kXRcRFZdl5wJURMQDcw+4+AJIk\n",
       "qWLNJv5ZwFM1y09RdNSbVGaeR5Hox1vcZBySJKkBzSb+24HrI2IFxY+A3y/LJElSF2s28Z8HnAO8\n",
       "laKfwO3Ap6sKSpIktUazif/9mflR4FNjBRHxEYpn+JIkqUs1lPgj4hLgIOD1EfFCds/Q9yzgWEz8\n",
       "kiR1tUZb/DcALwJeBaxid+LfDvxZhXFJkqQWaCjxZ+adwJ0RcSPwVGZuLVv+ATzQigAlSVJ1mp2k\n",
       "5zxgeUQcRtHyPx8790mS1PWaTfynAe8E3gJck5knAUdXFpUkSWqJZhP/nMzcCrwO+FpEzAH2qS4s\n",
       "SZLUCs0m/n+IiDXAXIpb/d8AbqoqKEmS1BpNJf7M/GPgN4BjM3MXcG5mXlBpZJIkqXKNvse/LDOX\n",
       "RsTnasrG/o5m5lkVxydJkirU6Hv83yv/rppg3eg0Y5EkSS3W6Hv8N5V/V7QkGkmS1FKN3urfVbM4\n",
       "yu6R+wBGM3NOJVFJkqSWaLTF/5+dASPirsx8cTNfGhHHAJdk5okRcQSwAtgFrKHoKOhjA0mSWqDZ\n",
       "1/maFhEXAMspXgUEuBxYkpnHU9xBOK3dMUmS1C/anviB+4E3sfsxwdGZubr8fAtwUgdikiSpL7Q9\n",
       "8WfmDcCOmqLafgKbgXntjUiSpP7RaOe+4ZrF549bHs3Mw5uIobbD4CCwcYoYhoClTXyPJEm9aHhs\n",
       "TJ0ayzJzaKKNG32P/8RmIprCXRFxQmauAk4Bbpts4/JAhmrLImIBMDzB5pIk9bqFmTlS78aN9uqv\n",
       "u+I6jPXcfx/FFL8DwD3A9RV+hyRJqtFoi78S5Q+I48rPPwIWdyIOSZL6TSd69UuSpA4x8UuS1EdM\n",
       "/JIk9RETvyRJfcTEL0lSHzHxS5LUR0z8kiT1ERO/JEl9xMQvSVIfMfFLktRHTPySJPURE78kSX3E\n",
       "xC9JUh8x8UuS1EdM/JIk9ZFndToAgIiYDXwKOBLYCrwjMx/obFSSJPWebmnxvwEYyMzjgPcDl3U4\n",
       "HkmSelK3JP5fA74OkJnfBV7a2XAkSepNXXGrH9gPeLxmeWdEzM7MXXXuPwdg3bp1lQfWLhvXPsyG\n",
       "B380rTo2r32E0YPmsXXjpmnV89jIT9i+6Ylp1VNFHcZiLMbSHXX0YixPPPIoDx/6MPvss8+0Yumk\n",
       "mpw3p5H9Zo2OjlYfTYMi4jLgO5l5Xbn8UGYeuodth4ClbQxPkqSZZllmDk20olta/HcApwLXRcSx\n",
       "wN172rA8kKHasoiYCzwFHAHsbFmUnTUMLOx0EC3k8c1svXx8vXxs4PHNZHOA+4G9M3NrvTt1S4t/\n",
       "Frt79QOcmZn3NVjHaGbOqjy4LuHxzWwe38zVy8cGHt9M18zxdUWLPzNHgXM6HYckSb2uW3r1S5Kk\n",
       "NjDxS5LUR3op8S/rdAAt5vHNbB7fzNXLxwYe30zX8PF1Rec+SZLUHr3U4pckSVMw8UuS1EdM/JIk\n",
       "9RETvyRJfcTEL0lSHzHxS5LUR7piyN4qRcR/Ab4DHJiZ2zodT1UiYl/g/wD7A9uA38/Mn3Q2qupE\n",
       "xDzgfwODwADw3sz8Tmejql5EvBF4c2ae3ulYpisiZrN7jo2twDsy84HORlW9iDgGuCQzT+x0LFWK\n",
       "iL2Aq4HDgLnAhzPzps5GVZ2ImAMsBxYBo8DZmfmDzkZVrYg4EPg+8KpG5rfpqRZ/ROwHXEYxU1+v\n",
       "eQfwT5l5AkWCvKDD8VTtj4D/m5mLgbcBn+xoNC0QEVcAHwF6ZcKQNwADmXkc8H6K//Z6SkRcQJE8\n",
       "5nY6lhY4HVifmccD/x34RIfjqdrrgF2Z+Qrgg8DFHY6nUuUPt08DTzS6b88k/nKGv08DFwJbOhxO\n",
       "5TJzLGlA8Qt9QwfDaYX/BfxV+XkvevDfIcX00+fQO4n/14CvA2Tmd4GXdjaclrgfeBO98++s1nXA\n",
       "ReXn2cCODsZSucz8O+Bd5eICeu//mZcCVwFrG91xRt7qj4i3A+ePK34Q+GJm3h0RMIP/Q93D8b0t\n",
       "M78fEbcBvwKc3P7IqjHF8c0HvgCc1/7IqjHJ8f1tRCzuQEitsh/weM3yzoiYnZm7OhVQ1TLzhohY\n",
       "0Ok4WiEznwCIiEGKHwEf6GxE1cvMnRGxAngj8OYOh1OZiHgbxd2aWyPiQhrMdz0zZG9E/Aj4cbl4\n",
       "LPDd8rZxz4nil83NmXlEp2OpUkT8KnAt8L7M/PtOx9MKZeJ/V2a+pdOxTFdEXAZ8JzOvK5cfysxD\n",
       "OxxW5crEf21mvrzTsVQtIg4FbgA+mZkrOhxOy0TEQcB3gV/KzBl/NzEiVlH0WxgFjgISOC0zH65n\n",
       "/xnZ4p9IZr5w7HNEDDODW8QTKX/V/Tgzv0DxTKenbstFxIsoWh2/lZn/2ul4VJc7gFOB6yLiWODu\n",
       "DsejBpTJ8Fbg3Zm5stPxVC0izgAOycyPUjw63FX+M+OVfb0AiIiVFI2JupI+9FDiH6c3bmM83WeB\n",
       "v46Is4A5wJkdjqdqH6HozX9l+ahmY2a+sbMhtcTYr/RecCPw6oi4o1zutWuyVq/8O6u1BJgHXBQR\n",
       "Y8/6T8nMXukcfT2womwd7wWcl5lbOxxTV+iZW/2SJGlqPdOrX5IkTc3EL0lSHzHxS5LUR0z8kiT1\n",
       "ERO/JEl9xMQvSVIfMfFLktRH/j8Psg+oUmeEOAAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f1c6510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z_mag = np.abs(z)\n",
    "z_arg = np.angle(z)\n",
    "\n",
    "plt.subplot(211)\n",
    "plt.hist(z_mag, 20)\n",
    "plt.ylabel('Histogram of |z|')\n",
    "plt.subplot(212)\n",
    "plt.hist(z_arg, 20)\n",
    "plt.ylabel('Histogram of arg(z)')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider two independent complex gaussian r.v. $w$ and $z$ both $\\mathcal{CN}(0, \\sigma^2)$. Let $w = |w|e^{j\\theta}$ and $p = |p|e^{j\\omega}$.\n",
    "\n",
    "**Q** What is the distribution of the phase of the product $p = wz$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": [
       "iVBORw0KGgoAAAANSUhEUgAAAesAAAFbCAYAAAD1OabUAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n",
       "AAALEgAACxIB0t1+/AAAEylJREFUeJzt3XuMpfV93/HPclmorQVHEfbSgLI0hG9dJchy0tomKSwK\n",
       "dUwVSmylaiN6sVM7F1PLbi0RexOZofJNdUljFDdyoQ5u6roqiF6QReoKUXCRbKeWJUJJvgRrJooV\n",
       "1tkSCJfYLHvpH3M2GS17YWbP8Pxm5vWSEOc855znfB8t7Huey5yz7fDhwwEAxnXa1AMAACcm1gAw\n",
       "OLEGgMGJNQAMbqhYV9XC1DOsJ9u3sdm+jWszb1ti+za6l7J9Q8U6yY1TD7DObN/GZvs2rs28bYnt\n",
       "2+hOun1nnOwJVfWGJB/v7iur6nVJbklyMMnzSf5Rd/9xVb0ryc8mOZDkw939hVObGwA44oR71lV1\n",
       "Q5Jbk5w1W/SrSf5pd1+Z5K4kv1hVr0nyniSXJfnxJB+rqu3rNzIAbC0nOwz+WJK3Jdk2u//3u/uh\n",
       "2e0zk3w7yd9I8mB3v9DdT89ec+l6DAsAW9G2k32CWVXtSvL57n7TimWXJbktyd9M8pYkP9jdH5g9\n",
       "9tkk/767713NIFV1VpLvJLk4y4fZN6PFJBdNPcQ6sn0b22bevs28bYnt28hOz/JO7tnd/fzxnnTS\n",
       "c9ZHq6q/l2RPkr/d3U9U1dNJdqx4yo4kT55kHQs5/gn1x1Y70wazOPUA68z2bWybefs287Yltm+j\n",
       "+05VHb3spu5eSFYZ66r6B1m+kGx3dx8J8leTfGS2Z3x2ktcmefhE65m9+cJR6/6+JI997nOfy86d\n",
       "O1czFgBsSHv37s11112XJBd39zeO97yXGuvDVXVakk8m+YMkd81+Avhf3X1TVd2S5EtZPge+p7v3\n",
       "r2Hmg0myc+fOXHDBBWt4OQBsWCc8/XvSWHf3Upav9E6S7z7Oc27L8jlsAGDORvtQFADgKGINAIMT\n",
       "awAYnFgDwODEGgAGJ9YAMDixBoDBiTUADE6sAWBwYg0AgxNrABicWAPA4MQaAAYn1gAwOLEGgMGd\n",
       "9PusAZjW/v37s7S0NPUYf27Xrl3Zvn371GNsKWI9R+/56I3Jd71y6jGSJJd9z8X56b/ztqnHAOZg\n",
       "aWkp77v7M9lx/nlTj5JnHt+XX73mZ3LJJZdMPcqWItZz9Pw527PtBy6YeowkyZPffHbqEYA52nH+\n",
       "eXnVhedPPQYTcc4aAAYn1gAwOLEGgMGJNQAMTqwBYHCuBgfgJTt44EAWFxenHuPPbZXf+RZrAF6y\n",
       "P9v3ZD65777seOLhqUfZUr/zLdYArIrf+X75OWcNAIMTawAYnMPgAMcw0pdnjHRBF9MQa4BjGOnL\n",
       "M/Y+1Nl5aU09BhMSa4DjGOVCqmce3zf1CEzMOWsAGJxYA8DgxBoABifWADA4sQaAwbkaHCYw0u/w\n",
       "JlvnyxBgoxJrmMBIv8O7lb4MATYqsWbLGGlvdnFxcZjf4QXGJ9ZsGSPtzfpEKmA1xJotZZS9WZ9I\n",
       "BayGq8EBYHBiDQCDO+lh8Kp6Q5KPd/eVVXVxktuTHErycJLru/twVb0ryc8mOZDkw939hXWcGQC2\n",
       "lBPuWVfVDUluTXLWbNGvJNnT3Zcn2Zbk2qrameQ9SS5L8uNJPlZVfmETAObkZIfBH0vytiyHOUle\n",
       "390PzG7fk+SqJH89yYPd/UJ3Pz17zaXrMSwAbEUnjHV335XlQ9tHbFtx+5kk5yY5J8mfHmM5ADAH\n",
       "q/3VrUMrbp+T5KkkTyfZsWL5jiRPnmglVbWQ5MZVvjcAbFaLVS/67IWbunshWX2sv15VV3T3/Umu\n",
       "TnJvkq8m+UhVnZXk7CSvzfLFZ8c1e/OFlcuqaleSxVXOAwCbwUXdvXS8B19qrA/P/v3+JLfOLiB7\n",
       "JMmds6vBb0nypSwfVt/T3ftPYWAAYIWTxnpW+stmt38/ye5jPOe2JLfNeTYAID5uFLa8gwcOZHFx\n",
       "nDNQvq4TXkysYYv7s31P5pP77suOJ054qcnLwtd1wrGJNTDMF5wAxybWwDBGOiQ/yhyQiDUwkJEO\n",
       "yfvOcUYi1sBQRjkk7zvHGYmvyASAwYk1AAxOrAFgcGINAIMTawAYnFgDwODEGgAGJ9YAMDixBoDB\n",
       "iTUADE6sAWBwYg0AgxNrABicWAPA4MQaAAbn+6xZV/v378/S0tLUYyRJFhcXpx4BYE3EmnW1tLSU\n",
       "9939mew4/7ypR8nehzo7L62pxwBYNbFm3e04/7y86sLzpx4jzzy+b+oRANbEOWsAGJxYA8DgxBoA\n",
       "BifWADA4sQaAwYk1AAxOrAFgcGINAIMTawAYnFgDwODEGgAGJ9YAMDixBoDBiTUADE6sAWBwYg0A\n",
       "gxNrABicWAPA4MQaAAZ3xmpfUFWnJbktySVJDiV5V5KDSW6f3X84yfXdfXh+YwLA1rWWPes3J3ll\n",
       "d/9okn+R5KNJbk6yp7svT7ItybXzGxEAtra1xPrbSc6tqm1Jzk2yP8kPdfcDs8fvSXLVnOYDgC1v\n",
       "1YfBkzyY5Owkv5fku5Nck+TyFY8/m+WIAwBzsJZY35Dkwe7+paq6IMl9Sc5c8fiOJE+daAVVtZDk\n",
       "xjW8NwBsRotVdfSym7p7IVlbrF+Z5OnZ7Sdn6/h6VV3R3fcnuTrJvSdawezNF1Yuq6pdSRbXMA8A\n",
       "bHQXdffS8R5cS6w/keQ3qupLWd6j/mCSryW5taq2J3kkyZ1rWC8AcAyrjnV3P5Xkrcd4aPcpTwMA\n",
       "vIgPRQGAwYk1AAxOrAFgcGINAIMTawAYnFgDwODEGgAGJ9YAMDixBoDBiTUADE6sAWBwYg0AgxNr\n",
       "ABicWAPA4MQaAAYn1gAwOLEGgMGJNQAMTqwBYHBiDQCDE2sAGJxYA8DgxBoABifWADA4sQaAwYk1\n",
       "AAzujKkHYP4OvnAg33p8bx599NGpR8ni4uLUIwBseGK9CT2zd19++/Cf5Pf+z91Tj5K9D3V2XlpT\n",
       "jwGwoYn1JrXj/PPyqgvPn3qMPPP4vqlHANjwnLMGgMGJNQAMTqwBYHBiDQCDE2sAGJxYA8DgxBoA\n",
       "BifWADA4sQaAwYk1AAxOrAFgcGINAIMTawAYnFgDwODEGgAGt6bvs66qDya5JsmZSX4tyYNJbk9y\n",
       "KMnDSa7v7sNzmhEAtrRV71lX1e4kb+ruy5LsTvJXktycZE93X55kW5Jr5zgjAGxpazkM/uYkv1NV\n",
       "/zXJ3Un+e5If6u4HZo/fk+SqOc0HAFveWg6Dn5fkwiQ/keW96ruzvDd9xLNJzj3RCqpqIcmNa3hv\n",
       "ANiMFqvq6GU3dfdCsrZY/78kv9vdB5I8WlXfSfI9Kx7fkeSpE61g9uYLK5dV1a4ki2uYBwA2uou6\n",
       "e+l4D67lMPj/TvKWJKmqv5zkFUnuraorZo9fneSB47wWAFilVe9Zd/cXquryqvpqlmP/7iRLSW6t\n",
       "qu1JHkly51ynBIAtbE2/utXdv3iMxbtPbRQA4Fh8KAoADE6sAWBwYg0AgxNrABicWAPA4MQaAAYn\n",
       "1gAwOLEGgMGJNQAMTqwBYHBiDQCDE2sAGJxYA8DgxBoABifWADA4sQaAwYk1AAxOrAFgcGINAIMT\n",
       "awAYnFgDwODEGgAGJ9YAMDixBoDBiTUADE6sAWBwYg0AgxNrABicWAPA4MQaAAYn1gAwOLEGgMGJ\n",
       "NQAMTqwBYHBiDQCDE2sAGJxYA8DgxBoABifWADA4sQaAwYk1AAxOrAFgcGINAIM7Y60vrKpXJ/la\n",
       "kh9LcijJ7bN/P5zk+u4+PI8BAWCrW9OedVWdmeTTSZ5Lsi3JryTZ092Xz+5fO7cJAWCLW+th8E8k\n",
       "+fUkj8/uv767H5jdvifJVac6GACwbNWxrqq3J9nX3V+cLdo2++eIZ5Oce+qjAQDJ2s5ZvyPJ4aq6\n",
       "Ksnrknw2yXkrHt+R5KkTraCqFpLcuIb3BoDNaLGqjl52U3cvJGuIdXdfceR2Vd2X5OeTfKKqruju\n",
       "+5NcneTek6xjIcnCymVVtSvJ4mrnAYBN4KLuXjreg2u+GnyFw0nen+TWqtqe5JEkd85hvQBATjHW\n",
       "3X3liru7T20UAOBYfCgKAAxOrAFgcGINAIMTawAYnFgDwODEGgAGJ9YAMDixBoDBiTUADE6sAWBw\n",
       "Yg0AgxNrABicWAPA4MQaAAYn1gAwOLEGgMGJNQAMTqwBYHBiDQCDE2sAGJxYA8DgxBoABifWADA4\n",
       "sQaAwYk1AAxOrAFgcGINAIMTawAYnFgDwODEGgAGJ9YAMDixBoDBiTUADE6sAWBwYg0AgxNrABic\n",
       "WAPA4MQaAAYn1gAwOLEGgMGJNQAMTqwBYHBiDQCDO2O1L6iqM5N8Jsn3JjkryYeT/G6S25McSvJw\n",
       "kuu7+/D8xgSArWste9bXJdnX3ZcneUuSTyW5Ocme2bJtSa6d34gAsLWtJdZ3JPnQite/kOT13f3A\n",
       "bNk9Sa6aw2wAQNZwGLy7n0uSqtqR5XD/cpJ/teIpzyY5dy7TAQCrj3WSVNWFSe5K8qnu/nxV/csV\n",
       "D+9I8tRJXr+Q5Ma1vDcAbEKLVXX0spu6eyFZ2wVmr0nyxSTv7u77Zou/XlVXdPf9Sa5Ocu+J1jF7\n",
       "84Wj1rsryeJq5wGATeCi7l463oNr2bPek+XD3B+qqiPnrt+b5Jaq2p7kkSR3rmG9AMAxrOWc9Xuz\n",
       "HOej7T7laQCAF/GhKAAwOLEGgMGJNQAMTqwBYHBiDQCDE2sAGJxYA8DgxBoABifWADA4sQaAwYk1\n",
       "AAxOrAFgcGINAIMTawAYnFgDwODEGgAGJ9YAMDixBoDBiTUADE6sAWBwYg0AgxNrABicWAPA4MQa\n",
       "AAYn1gAwOLEGgMGJNQAMTqwBYHBiDQCDE2sAGJxYA8DgxBoABifWADA4sQaAwYk1AAxOrAFgcGIN\n",
       "AIMTawAYnFgDwODEGgAGJ9YAMDixBoDBiTUADO6Mea2oqk5L8m+SXJrk+STv7O5vzGv9ALBVzXPP\n",
       "+ieTbO/uy5J8IMnNc1w3AGxZ84z1jyT5rSTp7q8k+eE5rhsAtqy5HQZPck6Sp1fcP1hVp3X3oZf4\n",
       "+tOTZO/evXMc6eX1ZC9l2588OfUYeeaJp3Lgu/5Snn/qmalHyZ8u/VFeeOY5s5jFLGaZu+f++Il8\n",
       "68Jv5RWveMXUo6zZiuadfqLnbTt8+PBc3rCqbk7y5e6+Y3b/D7v7wuM8dyHJjXN5YwDYnG7q7oVk\n",
       "vnvWDya5JskdVfXGJA8d74mzN19YuayqzkrynSQXJzk4x7lGspjkoqmHWEe2b2PbzNu3mbctsX0b\n",
       "2elJHktydnc/f7wnzXPPelv+4mrwJHlHdz+6ynUc7u5tcxloQLZvY7N9G9dm3rbE9m10L2X75rZn\n",
       "3d2Hk/zCvNYHACzzoSgAMDixBoDBjRbrm6YeYJ3Zvo3N9m1cm3nbEtu30Z10++Z2gRkAsD5G27MG\n",
       "AI4i1gAwOLEGgMGJNQAMTqwBYHBiDQCDm+cXecxNVf3VJF9O8uru3j/1PPNSVa9M8h+TvCrJ/iT/\n",
       "uLv/aNqp5qeqzk3yH5LsSLI9yT/v7i9PO9X8VdVbk/xUd1839SynqqpOy198pv/zSd7Z3d+Ydqr5\n",
       "q6o3JPl4d1859SzzVFVnJvlMku9NclaSD3f33dNONT9VdXqSW5NckuRwkp/v7v877VTzVVWvTvK1\n",
       "JD92ou/TGG7PuqrOSXJzlr+Ba7N5Z5Lf7u4rshy1GyaeZ97+WZL/2d27k7w9yacmnWYdVNUnk3w0\n",
       "yWb5UoGfTLK9uy9L8oEs/7+3qVTVDVn+C/+sqWdZB9cl2dfdlyd5S5Jfm3ieefuJJIe6+0eT/HKS\n",
       "j0w8z1zNftj6dJLnTvbcoWI9++auTyf5YJJvTzzO3HX3kb/ok+WfhJ+ccJz18K+T/NvZ7TOzCf8M\n",
       "s/xVsL+QzRPrH0nyW0nS3V9J8sPTjrMuHkvytmyeP7OV7kjyodnt05IcmHCWuevu/5bk52Z3d2Xz\n",
       "/Z35iSS/nuTxkz1xssPgVfVPkrzvqMV/kOQ/dfdDVZVs4P+5jrN9b+/ur1XVvUl+IMmbX/7J5uMk\n",
       "27czyW8mee/LP9l8nGD7/nNV7Z5gpPVyTpKnV9w/WFWndfehqQaat+6+q6p2TT3Heuju55KkqnZk\n",
       "Ody/NO1E89fdB6vq9iRvTfJTE48zN1X19iwfFfliVX0wJ+ndUB83WlW/n+Sbs7tvTPKV2SHVTaeW\n",
       "fxr5QndfPPUs81RVP5jk80ne393/Y+p51sMs1j/X3T899SynqqpuTvLl7r5jdv8Pu/vCiceau1ms\n",
       "P9/db5p6lnmrqguT3JXkU919+8TjrJuqek2SryR5bXdv+KN2VXV/ls/DH07yuiSd5Nru/taxnj/U\n",
       "BWbd/f1HblfVYjbwnuexzH56+mZ3/2aWz1FsqkNWVfXXsvzT/d/t7t+Zeh5ekgeTXJPkjqp6Y5KH\n",
       "Jp6HVZgF7ItJ3t3d9009z7xV1T9MckF3fyzLp9UOzf7Z8GbXLiVJquq+LO8AHDPUyWCxPso4u/zz\n",
       "8++SfLaqfibJ6UneMfE88/bRLF8FfsvsNMZT3f3WaUdaF0d+Gt4M/kuSv1VVD87ub7b/JlfaLH9m\n",
       "K+1Jcm6SD1XVkXPXV3f3ZrlA984kt8/2Qs9M8t7ufn7imSYx1GFwAODFhroaHAB4MbEGgMGJNQAM\n",
       "TqwBYHBiDQCDE2sAGJxYA8Dg/j/zD6F49cFsyAAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fa1d0d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Generate complex gaussian r.v. samples for w\n",
    "u = np.random.normal(loc = mu, scale = sd/np.sqrt(2), size = 1000)\n",
    "v = np.random.normal(loc = mu, scale = sd/np.sqrt(2), size = 1000)\n",
    "w = u + 1j*v\n",
    "p = w*z\n",
    "p_arg = np.angle(p)\n",
    "h = plt.hist(p_arg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The phase of the product still seems to be uniform over $[-\\pi, \\pi)$. The phase of the product $p$ is sum of the phase of $w$ and $z$, both of which are uniform. The pdf of the sum of two r.v. is given by the convolution of their respective pdfs. Following this notion, the pdf of $\\omega$ should have been convolution between two uniform distribution over $[-\\pi, \\pi)$ which leads to a triangular pdf over $[-2\\pi, 2\\pi)$. However, it is not the case as seen in the histogram above. What are we missing here?\n",
    "\n",
    "It must be noted that a complex exponential function is periodic with period of $2\\pi$, i.e., $e^{j\\theta} = e^{j(\\theta + k*2pi)}$ where $k$ is an integer. Hence the pdf of the phase $\\theta$ is periodic too. Since the phase $\\theta$ is periodic, the convolution between the phase pdfs should be a *periodic convolution* rather than *linear convolution*. A periodic convolution between two uniform pdfs results in another uniform pdf. This justifies the result obtained above."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}