{
 "metadata": {
  "name": ""
 },
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 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Fast matrix biquad\n",
      "==================\n",
      "\n",
      "This notebook shows the math behind a very fast NEON implementation of biquad IIR filters.\n",
      "One of the main primitives NEON is very highly optimized for is multiplying a matrix by\n",
      "a vector, especially of size 4.\n",
      "\n",
      "A biquad filter has two state variables (this is most apparent in transposed direct form II).\n",
      "For this implementation, the input vector consists of two input values and the two state\n",
      "variables. The output vector is two output values and the new state vector. The relationship\n",
      "is linear, ie the transformation from input to output vector is simply a matrix-vector\n",
      "multiplication. Each iteration computes two samples, using a single 4x4 matrix-vector multiply."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 117
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We start with a simple implementation of biquad for reference; we'll compare the error of the matrix approach against this scalar implementation. This is in direct form I."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def biquad(coefs, inp):\n",
      "    b0, b1, b2, a1, a2 = coefs\n",
      "    out = zeros(len(inp))\n",
      "    x1 = 0\n",
      "    x2 = 0\n",
      "    y1 = 0\n",
      "    y2 = 0\n",
      "    for i in range(len(inp)):\n",
      "        x = inp[i]\n",
      "        y = b0 * x + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2\n",
      "        out[i] = y\n",
      "        x2, x1 = x1, x\n",
      "        y2, y1 = y1, y\n",
      "    return out"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 118
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As a starting step, here's the same biquad in a 2x2 matrix formulation. This version is derived from\n",
      "transposed direct form II, with a little expansion of the effect _this_ input sample has on the _next_\n",
      "state vector (this is the B array below). The A matrix describes the evolution of the IIR state."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def bqmat(coefs, inp):\n",
      "    b0, b1, b2, a1, a2 = coefs\n",
      "    A = array([[-a1, 1], [-a2, 0]])\n",
      "    B = array([b1 - a1 * b0, b2 - a2 * b0])\n",
      "    y = array([0, 0])\n",
      "    out = np.zeros(len(inp))\n",
      "    for i in range(len(inp)):\n",
      "        x = inp[i]\n",
      "        out[i] = y[0] + x * b0\n",
      "        y = dot(A, y) + B * x\n",
      "    return out"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 119
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's test it out."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "impulse = zeros(100)\n",
      "impulse[10] = 1\n",
      "coefs = 1.0207, -1.7719, .9376, -1.7719, 0.9583  # peaking\n",
      "plot(biquad(coefs, impulse))\n",
      "plot(bqmat(coefs, impulse))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 120,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10cecce10>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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wIUaMGAHTNDF16lQkJydj0aJFAIDMzEyfTLIpmTAQVGcJSIVHsAEQUeDzKgAAYNSoURg1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\nxQAgIpIUA4CISFL/H9fdRioZk9ZlAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10ced4110>"
       ]
      }
     ],
     "prompt_number": 120
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(biquad(coefs, impulse) - bqmat(coefs, impulse))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 121,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10d3a9e50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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gaktCAi22duWK/EFm40bqPJUz/WNhSQM9/rj8ZQOUWl28WLlUk6QA4OvrC7PZ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\nYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10d390750>"
       ]
      }
     ],
     "prompt_number": 121
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note the scale of the error -- it's floating point rounding.\n",
      "\n",
      "Now for the 4x4 matrix approach. It's really four 2x2 matrices tiled. The top\n",
      "left corner is the influence of the two input samples on the two output samples;\n",
      "it's the first two values of the biquad's impulse response, banded. The top\n",
      "right corner is the contribution of the state vector to the output. The bottom\n",
      "left is the effect of the input on the state, and the bottom right is the\n",
      "evolution of the IIR. Since each iteration is two samples, this matrix is\n",
      "simply the square of the A matrix above.\n",
      "\n",
      "Note that each iteration processes _two_ samples. The matrix-vector product\n",
      "is 16 multiply-accumulates, so it amortizes to 8 per sample. This compares\n",
      "well to the 5 multiply-accumulates of the direct form, but organized for extremely\n",
      "fast evaluation in SIMD, and NEON in particular."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def biquadm(coefs, inp):\n",
      "    b0, b1, b2, a1, a2 = coefs\n",
      "    c1 = b1 - a1 * b0\n",
      "    c2 = b2 - a2 * b0\n",
      "    A = array([[b0, 0, 1, 0],\n",
      "               [c1, b0, -a1, 1],\n",
      "               [-a1 * c1 + c2, c1, -a2 + a1*a1, -a1],\n",
      "               [-a2 * c1, c2, a1*a2, -a2]])\n",
      "    out = np.zeros(len(inp))\n",
      "    y = zeros(4)\n",
      "    for i in range(0, len(inp), 2):\n",
      "        y[0:2] = inp[i:i+2]\n",
      "        y = dot(A, y)\n",
      "        out[i:i+2] = y[0:2]\n",
      "    return out"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 122
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Again, let's test it out."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(biquad(coefs, impulse))\n",
      "plot(biquadm(coefs, impulse))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 123,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10d3b9510>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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wIUaMGAHTNDF16lQkJydj0aJFAIDMzEyfTLIpmTAQVGcJSIVHsAEQUeDzKgAAYNSoURg1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\nxQAgIpIUA4CISFL/H9fdRioZk9ZlAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10d3b94d0>"
       ]
      }
     ],
     "prompt_number": 123
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The error is similar to that of the 2x2 matrix version, but not quite the same."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(biquadm(coefs, delta) - biquad(coefs, delta))\n",
      "plot(biquadm(coefs, delta) - bqmat(coefs, delta))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 124,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10d70b550>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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NsEbtoUOGU/V+3vlzBP8UjDXJa/BiC4E7rIC3AIQ+vTHG8H78+0i6loS9I/bi\nmSrGM4ldu8bTHphKANfPpx/2vLYHg34fhH1X9mFuxFxUta9q9DOhobwVdu0aTzAoxuOCx5j+z3TM\nTZiLF/xfwPG3jsOjtkeZ4zZtAgb2ccQ77d7BO+3ewbFrx7AocRFCfg5BsEswXgp4CT2b9YT7M8IG\nCHTdQH36iKu/OTIyePqJNWuMH5d2Jw3r/12P89nncSnnEi7cvoArd66go0dHdGvSDb08e+n9/2uX\nWITzJ07g+7vfY+gfQzGy1UjM7jNbnsqLDh2MsTVr1rDXX3+9+Ptly5axsWPHljqmX79+bF+J9ek9\nevRgR44cKVOWxKpY1eWcy6zdwnas/6r+7PbD2yyvII/tu7KPTdFOYa7furIBqwawE9dOGC1j0iTG\nvvjCcnX89VfGXn5ZWhmFRYWs2exm7HDGYb3vnznDmKeneWXlPMxhjjMcWfodE82FEiZPZmzCBNPH\naS9pmdccL8HpJ+7cYaxGDeOzNvZf2c8aftNQVCsvL4+x6tUZu3vXvOOLiorY+G3jWeufWrPbD2+b\n9Zk1axjr29f8Ot19dJe9uOZFFrwgmGXlZpk8fsAAxpYuNb/8knS/J5G/RbIL2RcMHldYyFuS58+X\nfe9h/kO28uRKNmj1IFY3ti7zmevD3tr0Flt9arVZ9U9NZaxePcbyxWcmMembb3hLUp9H+Y/Y76d+\nZ72X9WZOsU5s5PqR7Nv937I/k/9kSVeT2OOCxybLP3GCMVdXxh4/5tcj7U7paVJS7p2SWgDm9rWy\np/p+DX3O+T+RUqojK3t7ILgVoO+B8nDGYXzY/kN81OGj4p+lg0cHdPDogPEdxmP+kfnouawnonyi\nML/ffNhXKn2ZFyUuwq9561GrJpC0snTZdho7DG05FC+0eEHw02xJciwG06ZqUbNyTbRx0d+iMbQG\nQJ/aVWvjpYCX8OPhHzGtxzSzPhMYCMSZMQ779d9f47NOn5W5zqYcPszTNhibtdHeoz2GtBiCD7Z9\ngCX9lwgq38GB/wyJiUBXM9akfbXnK/z3wn+xa/gu1Klax6xz/P23eWXr1KpSC6sGrcInOz5B9Npo\nbHtlm9Hr1rs33yNA6L612y5sw/B1w8v8nuhz5Ajg5AR4epZ9r6p9VUS3jEZ0y2gUsSKczDqJnZd2\nYtmJZXhj0xtoXLsxGtd58p+wdpXa6NK4C7o37Q5PR080bqxBkyb8OnXvLuxnMNeKFTz9Q0kns05i\nUdIirDgRv6cAAAAgAElEQVS5AgENAjAqeBTWDVlXZhadOQIC+ID82rXAsGFVBbeCjJEUANzc3JCW\nllb8fVpaGtyfmsP19DHp6elwc3PTW17z1JrFf2/sF4AmfgFSqifJd98DLw7Sn81veo/paNlA/+7n\n1Ryq4YP2H+DNNm/ixTUv4pV1r2D5gOWwq2QHAFhwZAFm/DMDDa/OQtfODujUuvTn7+Xdw6yDszBt\n7zR81e0rPN/8eUGDmjpydAH9kvgLXg9+3eD5he4D8G7ou+gU1wkTu0w06xdB1wVkzIG0AziffR4v\nB75sfkX+59Ah4FnjqYIAAFO7T0XA/ADEn4/Hc17PCTqHbkWwqZv0r8d+xZLjS3Bg1AHUrV7X7PL3\n7AF+/llQlaDRaDC9x3T0WdEHE3dNxIyeMwwe27v3k7QE5qTKzribgWl7p2HDmQ34/YXf0aVxF5Of\n2bSJr341pZKmEoIaBiGoYRA+aP8BCooKkHg1sdRq8+v3r0N7WYspe6aAMYbaVWvjViQwaDfQsMSs\nMjuNHepUrVP85VjVkf9ZzRFV7atCA/N+565dAy46AUlVH2Fj/GWk5qTi7K2zuPv4Ll5r9RoOjjoI\nTyc9kU2gceP4/gLDhgFarRZarVZymQCgYU8/ngtQUFCA5s2bY+fOnXB1dUW7du2wcuXKUmMAW7Zs\nwbx587BlyxYcPHgQ77//Pg4eLLtKUcrO9pbg5wf8+Sf/U6yH+Q8RtSoKrrVcsfj5xfj12K+YrJ0M\n7WtaDI9shpgYvgHH0xhj+OvsX/hS+yXsNHb4utvXeM7rOUGB4N49Psvo/n1xSbWyH2aj2exmuDju\nIpyqOek9ZuJE/pQ7ScBi34gVERjsPxgjg01vf1RUxDfTyMjgG63o0/e3voj0idQ7RdWUqCj+ZPvC\nC6aP3X5hO17f9DpOvX0KtaqYv/nx0qV8fvuqVYaP2XVpF4b+MRTaV7Xwq2/+f7jsbB6As7PFzT2/\n+eAm2vzcBt/3/h4D/AYYPM7Xlz/lGks0l5WbhRn/zMDSE0sxKngUPu74MepVr2dWPYKD+X6+nTsL\n/QkMY4zh8p3LeJj/EKmpwPBXAe3uJ0GsoKgAOY9ykPMoB7cf3eZ/PryN249uC9rTIuEw3/2taycH\nNK7dGE3qNEEzx2YIdA4sfuiTQ2Ehnxm3cmXZWWuS7p2iO4/+Z8uWLczHx4d5enqymJgYxhhjCxYs\nYAsWLCg+5p133mGenp4sMDCQHT16VG85MlRFVh068LS+Ut3Pu8+6Lu7Kwn4NY67furIzN88wxhjz\n9mbs33+Nf7awqJCtOb2G+f/gz9r/0p7tvLhT0Llr12bs1i1x9Z5zcA4bunao0WNefpmxxYuFlRt/\nLp4Fzg80e2ZN27aG/x2OZh5lbt+6sYf5D4VVgvGV2A0aMHblivmfGbF+BHtn8zuCznP6NJ+lY/D9\n66dZg5kN2O5LuwWVyxhjGzcy1rOn4I+Vcij9EKv/f/XZvzcM/2d87z3Gpk3T/55udo9TrBN7b8t7\n7Oq9q4LOf+UKY3XrWraPnjHGWraU5/e5pKIi/m9r4JYmu2+/ZWyonl9JKfdO1dx11RYA+vblv2By\nuPf4Hhu1YVSplAjPPMNYdrZ5ny8oLGDLjy9nXnO8WLdfu7F/Lpv3P9nfn7Hjx4XXt6ioiAXODzQZ\ncDp3ZmzXLuFl+//gz+LPxZt1/KhRjM2fr/+9AasGsO8PfC+sAv9z6RJjLi7C8q5nP8hmTb9vyn5I\nMDBvVI+CAsZq1tQfiM/ePMsaf9eYLTkmLnn9Rx8x9tVXoj5ayuKkxcx5pjPbdn6b3vc3b2asS5fS\nr91+eJt9uetLVje2Lhu9aTS7kiMgkpYwf770yQrmmDyZsXHj5C3zwAHGmjcXv2+BULdvM1anDmMZ\nGaVfpwBgAa+8wmfSWMLDhzxXj9D/OPmF+WxR4iLW+LvG7LnlzxmcnaPTq5e4OdAH0w6ypt83NTmn\nvlEjxi4Yntxh0PLjy1mnuE5mtQK+/56xMWPKvn4y6yRznunM7ueJS6a0cqW4dRgXsy+yRt81YguP\nLjT7M126MPbf/5Z+LSE9gTX8pqGgcp7Wti1je/aI/ngpuy/tZi7fuLAp2ill/t1zc3kQu3OHP8xM\n+3saq/d/9dir6141OrvHHBERjK1aJakIsyQn85k0hcKWiRg1dqw8AViIMWMYmzix9GsUACxg3DjG\nZs2yTNmXLzPm5ib+848LHrMfE35kbt+6sedXPs+OX9P/mD9yJGM//yys7ILCAtZuYTv205GfjB6X\nn89Y5cp8appQ+YX5zGuOF9Ne0po8dudOxjp1Kvt69NpoFvtPrPCT/8/HHzP29dfiPnvu1jnm9q2b\n2U/uEyaU/qXdem4rq/d/9diGf8Vvu3X3Lp9i+lB475dBGXczWKe4TqzL4i5s/uH57MzNM6yoqIjd\nfXSXBQ3ezKJ++IA5z3Rm0WujjXYZmSs3l7FatRjL0b/GUHYBAYz9/bc8ZeXn8y7Ec+fkKc9c587x\nRaQlSbl3Ui4gA5ycLLevqNQ0EJXtKuPttm/j3LvnENYkDL2W9cJXe74qc5yYmUBzDs1BdYfqeL21\n8Zy2GRl8IVvlykYP08u+kj0mdJqAqXunmjxWNxOo5BjXyayT2HFxB94OeVv4yXVlnORli+Hl5IUd\nw3fg0x2f4tMdnyL7YbbR47t3B3bt4n9fc3oNXl3/KjZEb0BU8yhxFQBfeNSmDVDV+FouQVxruWLX\n8F14Pfh1HEg/gB5Le6Dhtw3h8q0LcgO/wdWLdbBz+E6sHLQSzes1l3y+HTt42mxDA/xyGzIEWL1a\nnrJ27ACaNuX7VliTlxcwcKCMBUoMSLJRUVUYY4zNmcPYO8LG+8z211+M/S87hiyycrOY80xnlpCe\nUOr1n37ifejmupB9gdWNrcvO3TL9WKPV8mX2Ypm7Zy9jPMWubrC2oLCAtf25Lfv5iMCmzVPc3Rm7\nKC3jM0u7k8be3Pgmqxtbl03ePdlguoz79/mCs98S1zHnmc4GW2xCfPaZuHTZQhQVFbHU26nsQd4D\nduKEtH169Rk+nP+eWcvZs3zBmRxp0qOj5Um1Igcp905qARjg5MSn11mC3GkgGtRogFm9Z2HUxlGl\ndoMSshiMMYY3N72JTzp+Ai8n0481QtcAPM3BzgGfdPwEU/82rxVw4gT/++xDs1Gzck2TLRRjbt8G\n7tyRVn8AcH/GHT9F/oSENxJwMeciAuYH6N2IpXp1oFnvzRizZTS2vLQFgc6B0k4M4QvAxNBoNGhc\npzGqOVRDy5bA48fy7RKWl8cTqMn6NGuCtzfg6sqvnRS3bvHduuTMXKoUCgAGWDoAyJ0KaWjLoWhU\nuxFi/3mSkdXcLiDGGOYlzEPOoxz8p/1/zDqf1AAAACOCRyDpWhISryYaPU7XDXQh+wJi9sbg58if\nRS2O09HlFzJnYZM5mjk2w5L+SzAvYh6iVkZh7qG5xfOy7zy6g6XHl+Ji4Aj0ubMRrV1amyjNtIcP\n+eri9uanVZJM7l3Cdu/mmTMNrAm1mBdflN4NtHw533dY3yLR8oYCgAGWDADXr8ufCE6j0WB+3/mY\nkzAHyTf4TirmpITefWk3Oi3uhPlH5mNJ/yVmp1OQIwBUta+KcaHjMDdhrtHjAgKAEycZ3tj0BiZ0\nmmBWC8WYkyd5iga5RTWPwoFRB7AoaRF6L++Ndgvbwf07dyw9vhTftl2PlO2hpgsxQ0IC0LIlTxds\nTf368cWRcvjjD56ozdpefJH/DAUF4j7PGPDLL/Lm/VeSpFQQFZklB4Fv3uSrH+XmUdsDX3f7Gs+v\neh4hriEAA+6Ga7DkyPN4pU3p3EL70/bji91f4MqdK5jcdTKiW0YLWrl4+TIweLD0Or/W6jX4zPXB\n7OdmG8x8GRgITPzjV7jm5WLcs+Mkn1PKALApnk6eODDqAJYeXwrfer541v1ZVLGvgvx8YHw07z6o\na36mB7327JF31ay5+vbl+91mZEh7ci8sBNav56k4rK1ZM6BJE94CCQ8X/vnDh3kLzNLdb9ZCLQAD\nHB0t1wK4cQOoX98yZb/Z5k3M6jULUT5RiGoehbp3emLWgVlotaAV1v+7HkcyjyBiRQSG/TEMw1oO\nQ8o7KXgp8CXBy9blaAEAfPyiR7MeWHXKcK4EL598ZHhPwTc9ZgtO+KbPiROWCwAAzwc1OmQ0ujbp\niir2VQDwVA2dOvEbj1S7d1susZkxVavyTVaMpbUwx969vHXatKk89RJKSjfQokX86V+u7kPFyTUS\nLZWKqsIY4/N87ezkXTii07o13zTcGrp2ZWz79iK28d+NLGh+EHP91pX9kPADe5T/SHSZhYWMVa0q\n34b2W85uYW1/bmvw/bjEOFb9rR7shPEM22YpKuJzz8WmyJDim2/4Rj1S6GYUmZtiWm7bt/P/v1KM\nHcvY1Kny1EeM9HS++fxt8zJuF7t3T/9KXKVJuXdWlDgmO3t73sd65478ZVuyBfA0d3cgM1ODyOaR\nSBqdhLT/pGFM2zHFT6ZiXL8O1Kol34b2vTx74WruVZzIOlHmvcKiQsT8E4O2D74wmRnUHJcv8wRz\nTvrz21lUyfUAYu3fz7vEapmfj05W3boBmZnAv/+K+3xREe+DV6L/X8fNjY9nLFgg7HNr1vCuN1dX\ny9RLCRQAjLDUOMDNm9YLACVnAmk0Gkl7DOgI2QfAHHaV7DCi1QgsSlxU5r3Vp1ejYc2G6O7ZRfIG\n64Dlu3+MCQri//ZS9mretQvo0UO+OgllZwdERwO//Sbu84cOAXXqmN7BzNI++ohnIH1sZuJPxnjA\nqCiDvzoUAIywxEyg+/f5nzXK7q9uEXJuDq8jV/9/SSNajcCKkyvwqOBR8WtFrAjT9k7DF12+QJs2\nGiQlST+PJQeATalUCQgLk9YK2LlTmf7/kl56iaeHFpOBeO1aZZ/+dQID+deKFeYdv24d3wK1Xz/L\n1svaKAAYYYmB4Bs3gHrmpUmXhRw7gz3NEgGgqWNTBLsEY/2/64tfW5eyDjUcaiC8WThatwaSksTd\ndEqy1BRQc3XvztMIiHHnDnD6tHXn/+vTpg3vIk1IEPa5u3f5/ghCdxezlPHjgZkzebeUMXl5wCef\n8F2/7ORL8a8KFACMsEQLwJr9/4B8m8OXZIkAAABvtH4D78e/j45xHdExriPe3vw2JnaZCI1GAxcX\nPpOmxOZyoijZBQTwqZSbNwP5+cI/+/fffAczOfP/iKHR8J2pzH161vnxR76YzNvbMvUSqnt3oFo1\n/u9hzPz5vM5ipo2qHa0DMMISAcCa/f+A5bqAevWSt0wAeMH/BTRzbFa8I1N1h+oIdnmyYKJ1a74C\ntlEjceU/fgxcuqRs/3PjxvxmsnMn8Jyw3SWxc6ey/f8lvfQS0KEDEBNj3oK0+/eB776TPgguJ43m\nSSvA0JaUt28D06apq95yEt0CyM7ORnh4OHx8fNCrVy/k5OToPa5JkyYIDAxEcHAw2rVrJ7qiSrDE\nILC1u4CcnXkQy8szfay5LNUC0Gg0CHENQcdGHdGxUcdSN3+AL55LNJ41wqiUFL7xeBXxE6BkITYr\n5a5dyvf/63h58SfimTPNO/7nn/kMmhYtLFsvoV54ge/r+/Sm7joxMUD//nzldUUkOgDMmDED4eHh\nOHv2LHr06IEZM/RvLK3RaKDVapGUlIQEoZ2GCqsIXUB2djwIZGbKUx5jPAA0aSJPeULoxgHEUnIA\nuKTBg4ENG4QF5evXgStXjO/La20xMcC8eaZbmI8e8Rvs559bp15C2NvzwPrrr8AHHzwZD8jP5ykf\n4uKAKVMUraJFiQ4AGzduxKuvvgoAePXVV7F+/XqDxzIVbfYuhKUGga0ZAAB5u4FycnjTuU4decoT\nQtcFJJbS/f867u78SXjbNvM/s3s30KULv2GpRePGwBtvAF9+afy4uDj+b2eJ9CdycHfnq5MPHwZe\nfpkPVPv58VZafDzg4qJ0DS1HdADIysqC8/9SWjo7OyMrK0vvcRqNBj179kRISAgWLlwo9nSKqAhj\nAIC8M4HkXgMgROPGPA/LtWviPq+WAAAI7wZSev6/IRMm8EHUE2XX8AHgM5diY4GJE61bL6EcHZ8E\n5MWLecqH7duBtm2VrZelGX2eCA8PxzU9v23Tpk0r9b1GozGYnnffvn1wcXHBjRs3EB4eDl9fX3RW\nIpOVCJbqArLmGAAgbwvAUv3/5tBonnQD9ekj7LN5ecDBgzyVrxoMGgR88QXvHjE1q6ewkKdhHjvW\nOnUTonZt/nN89BGvY8nbwL17fKC7f38gVJ5EqBZVrZr4BW7lldEAsH37doPvOTs749q1a2jYsCGu\nXr2KBgbyG7v8r/1Uv359DBgwAAkJCQYDwOTJk4v/HhYWhrCwMBPVtyxLDQJbuwUg51RQJQMA8GQg\nWGgAOHCAz76x9rU3xMUFaNWKdzH072/82Lg4Puai1oHIN9/kYwFjxvBg4OrKZ/307cvXXHz/vdI1\nrFi0Wi20Wq0sZYnuUYyKisKSJUvwySefYMmSJeiv53/xgwcPUFhYiFq1auH+/fvYtm0bJk2aZLDM\nkgFADSrSGIBc4+9KB4DWrXkueaH++1+gd2/56yPFkCHA778bDwC5ucCkScDGjaWfrtXEwYGnqI6N\n5UFqxAjg2DE+42r+fPXWu7x6+uF4ioRRatFjAJ9++im2b98OHx8f7Nq1C59++ikAIDMzE3379gUA\nXLt2DZ07d0arVq0QGhqKfv36oZclJpBbiK4LSM4xbKXGAOTqAkpNVWYGkI7YgeD4eOHz7i1t4EBg\nyxb+f8KQ//s/3vcfEmK9eonRoAHw7bfAqVO8uy0ggM+iqTBpkysoDVPJFB2NRqPK2ULVqvFNPOTI\nfJmfz8t5/Ni6vxiXLvEcNJcvSy8rKIhPmVNqRkdREe93vnLF/C35srKA5s1568vBwbL1E2r8eL7P\n7p9/ln1SzsjgXShJSeIXv5GKT8q9k+KzCXIOBN+8ycuz9lORqytw9arpnCemMMZvVp6e8tRLjEqV\neN+5kPUA27fzNMZqu/kDwNSpvFW1qGwiVEycyPvX6eZPLIUCgAlyDgQr0f8P8JWvjo58MZEU167x\nLKbP6N+50WqCg4UFADX2/+tUqcJz6nz6KXD2LH8tLw/44Qdg61b+OiGWQgHABDkHgpXo/9eRYybQ\nhQvKPv3rCBkHKCri87vVGgAAwN8fmDyZ59eJi+PdVZs28XGL2rWVrh2pyCgAmCBnF5ASawB05FgM\ndv48zwGjtJAQPq3TnG7P48f5qmWl9p811zvv8H+jpUv5V3w87+oixJJUtLBcneQOAEq1AOSYCaSW\nFkCLFvzmf+IEH5Q2Rs3dPyVpNHzTEUKsiVoAJlSEMQBAni4gtbQANBq+ktac9QDx8eUjABCiBAoA\nJsg9C0jJFkBFCQCAeQHg7l3g6FE+BZYQUhYFABPkHARWegygonQBATy3TE4O8O+/ho9ZtoznrLfW\n/suElDcUAEyoSGMAUloA2dk8KZlSAexplSrxlbSGWgFFRcDcucC4cdatFyHlCQUAEypKAHBz4y0A\nsYutdU//asrrYqwbaNs2nmWzSxfr1omQ8oQCgAlyDgIrOQZQsyZfCWtg506T1NT/r9O5M2/VXLxY\n9r3Zs4H33lNXwCJEbSgAmCBXC6CoiOcUqltXelliSekGUmMAsLPjmTT//LP062fO8IViw4YpUy9C\nygsKACbINQick8MHIytXll6WWLpuIDHUNABckr5xgLlz+VaFpjZaIcTWUQAw4Zln+OYW+fnSylGy\n/1+norUAAKB7d/7E/9VXwL59/DqvWAG8/bbSNSNE/SgAmFCpEk8lILbvXEfJ/n8dKQFArS2AypX5\nYq+7d4F33+WtnD59+J+EEOMoFYQZdAPBUm7gSq4B0HF35zl0hMrN5Zt7u7rKXyc5tGvHvwDeXVel\nirL1IaS8EN0CWLNmDVq0aAE7OzskGknNGB8fD19fX3h7eyM2Nlbs6RQlx0CwGrqAmjfn3SVCXbgA\nNGtWPnZ3cnKihV+EmEv0r3RAQADWrVuHLkYmWhcWFmLs2LGIj49HcnIyVq5ciZSUFLGnVIwcA8Fq\nCAD+/kBysvC1AGrt/iGESCM6APj6+sLHx8foMQkJCfDy8kKTJk3g4OCA6OhobNiwQewpFSNHC0AN\nYwD16gH29nyLRCHUOgBMCJHGoo36jIwMeHh4FH/v7u6ODLl2J7eievWMb9xtDjWMAQBPWgFCUAuA\nkIrJ6CBweHg4rl27Vub1mJgYREZGmixcI3AZ5uTJk4v/HhYWhjCVpHF0dhb+1Pw0NXQBAYCfH5CS\nwqdPmuv8eZ52gRCiPK1WC61WK0tZRgPA9u3bJRXu5uaGtLS04u/T0tLg7u5u8PiSAUBNGjTgN0Ep\n1BIAxLQAqAuIEPV4+uF4ypQposuSpQuIGRhVDAkJwblz55Camoq8vDysXr0aUVFRcpzSqpydpW+o\nfv26OgKArgVgrocPed0bNbJcnQghyhA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IITaKAgAhhNio/wfXy5BMpuKmKwAAAABJRU5E\nrkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10d70b5d0>"
       ]
      }
     ],
     "prompt_number": 124
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 124
    }
   ],
   "metadata": {}
  }
 ]
}