{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as pt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 2048\n",
      "\n",
      "spikes = np.zeros(n)\n",
      "spikes[730] = 1\n",
      "spikes[1512] = 0.3\n",
      "pt.plot(spikes)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "[<matplotlib.lines.Line2D at 0x36682d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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lRVmrTUNDgzZt2qTW1lZt2bJFkvTGG29o69at2rBhg77whS/ozJkzpfE//vGPFYvF1NjY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\nA4AB/wcAR43qoeflkgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x359e0d0>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pattern = np.zeros_like(spikes)\n",
      "pattern[n/2-100:n/2+100] = np.sin(np.linspace(-pi, pi, 200))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "-c:2: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pt.plot(np.convolve(pattern, spikes))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "[<matplotlib.lines.Line2D at 0x3f72910>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x3a64050>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Now with the FFT"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spikes_hat = np.fft.ifft(spikes)\n",
      "pattern_hat = np.fft.ifft(pattern)\n",
      "\n",
      "convolved = np.fft.fft(spikes_hat * pattern_hat)\n",
      "pt.plot(convolved.real)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "[<matplotlib.lines.Line2D at 0x3ec34d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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tbyEtLS1g/fT0dKxevRrp6ekoKCjAyy+/HPSQGhFNDLdO4F++DNx2m/wLk6R9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\nwUJERIr6f3WGK7hpnIsHAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3a52890>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    }
   ],
   "metadata": {}
  }
 ]
}