jedludlow/ipython_fft_example.ipynb

## ipython_fft_example.ipynb
{
 "metadata": {
  "name": "",
  "signature": "sha256:1c8a707035342477c2cad874c9553bf896070c3c12617eb0eb4e9f6d312accc1"
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 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Working with Numpy FFT Results: Scaling and Folding"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Imports\n",
      "-------"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Create a Test Signal\n",
      "--------------------\n",
      "$f_s$ is the sampling frequency, while $f$ is a base frequency for the signal content. We create a signal that contains components at a couple of multiples of this base frequency. Note the amplitudes here since we will be trying to extract those correctly from the FFT later."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_s = 50.0 # Hz\n",
      "f = 1.0 # Hz\n",
      "time = np.arange(0.0, 3.0, 1/f_s)\n",
      "x = 5 * np.sin(2 * np.pi * f * time) + 2 * np.sin(10 * 2 * np.pi * f * time)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(time, x)\n",
      "plt.xlabel(\"Time (sec)\")\n",
      "plt.ylabel(\"x\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "<matplotlib.text.Text at 0x6e22b10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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h+x+jcDBJE3AXkjZaC3wFObzk7xGD8gySolqTv/7fgG35\n99cBf5p/vxk5zOT5wHepKIqiRI417XRcBb/zcsSYKIqiKAnjL4H3VPD77kYGlBVFURRFURRFURRF\nURRFURRFURRFURRFURRFURRFURRFUeLE/wdcSEaGo9iOUgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x6e0ae30>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Compute the FFT\n",
      "---------------\n",
      "The FFT and a matching vector of frequencies"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fft_x = np.fft.fft(x)\n",
      "n = len(fft_x)\n",
      "freq = np.fft.fftfreq(n, 1/f_s)\n",
      "print n\n",
      "print freq"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "150\n",
        "[  0.           0.33333333   0.66666667   1.           1.33333333\n",
        "   1.66666667   2.           2.33333333   2.66666667   3.           3.33333333\n",
        "   3.66666667   4.           4.33333333   4.66666667   5.           5.33333333\n",
        "   5.66666667   6.           6.33333333   6.66666667   7.           7.33333333\n",
        "   7.66666667   8.           8.33333333   8.66666667   9.           9.33333333\n",
        "   9.66666667  10.          10.33333333  10.66666667  11.          11.33333333\n",
        "  11.66666667  12.          12.33333333  12.66666667  13.          13.33333333\n",
        "  13.66666667  14.          14.33333333  14.66666667  15.          15.33333333\n",
        "  15.66666667  16.          16.33333333  16.66666667  17.          17.33333333\n",
        "  17.66666667  18.          18.33333333  18.66666667  19.          19.33333333\n",
        "  19.66666667  20.          20.33333333  20.66666667  21.          21.33333333\n",
        "  21.66666667  22.          22.33333333  22.66666667  23.          23.33333333\n",
        "  23.66666667  24.          24.33333333  24.66666667 -25.         -24.66666667\n",
        " -24.33333333 -24.         -23.66666667 -23.33333333 -23.         -22.66666667\n",
        " -22.33333333 -22.         -21.66666667 -21.33333333 -21.         -20.66666667\n",
        " -20.33333333 -20.         -19.66666667 -19.33333333 -19.         -18.66666667\n",
        " -18.33333333 -18.         -17.66666667 -17.33333333 -17.         -16.66666667\n",
        " -16.33333333 -16.         -15.66666667 -15.33333333 -15.         -14.66666667\n",
        " -14.33333333 -14.         -13.66666667 -13.33333333 -13.         -12.66666667\n",
        " -12.33333333 -12.         -11.66666667 -11.33333333 -11.         -10.66666667\n",
        " -10.33333333 -10.          -9.66666667  -9.33333333  -9.          -8.66666667\n",
        "  -8.33333333  -8.          -7.66666667  -7.33333333  -7.          -6.66666667\n",
        "  -6.33333333  -6.          -5.66666667  -5.33333333  -5.          -4.66666667\n",
        "  -4.33333333  -4.          -3.66666667  -3.33333333  -3.          -2.66666667\n",
        "  -2.33333333  -2.          -1.66666667  -1.33333333  -1.          -0.66666667\n",
        "  -0.33333333]\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(np.abs(fft_x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "[<matplotlib.lines.Line2D at 0x70986f0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9jPngF6kV8/FeO0FrIbOCCPtUll7G/LVWpFbMx7vFpZdBhH0qv88+5kpHpFbMx/tcLb2c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\nIiIiIuH7f+nz76hXMY8kAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x6e31110>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Swap Half Spaces\n",
      "----------------\n",
      "Note that frequencies in the FFT and the `freq` vector go from zero to some larger positive number then from a large negative number back toward zero. We can swap that so that the DC component is in the center of the vector while maintaining a two-sided spectrum."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fft_x_shifted = np.fft.fftshift(fft_x)\n",
      "freq_shifted = np.fft.fftshift(freq)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(freq_shifted, np.abs(fft_x_shifted))\n",
      "plt.xlabel(\"Frequency (Hz)\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "<matplotlib.text.Text at 0x70a2b50>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXgAAAEPCAYAAABIut/fAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x707c690>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Fold Negative Frequencies and Scale\n",
      "------------------------------\n",
      "It's actually more common to look at just the first half of the unshifted FFT and frequency vectors and fold all the amplitude information into the positive frequencies. Furthermore, to get ampltude right, we must normalize by the length of the original FFT. Note the factor of $2/n$ in the following which accomplishes both the folding and scaling."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "half_n = np.ceil(n/2.0)\n",
      "fft_x_half = (2.0 / n) * fft_x[:half_n]\n",
      "freq_half = freq[:half_n]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(freq_half, np.abs(fft_x_half))\n",
      "plt.xlabel(\"Frequency (Hz)\")\n",
      "plt.ylabel(\"Amplitude\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "<matplotlib.text.Text at 0x6edc150>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x70daf50>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now the spectrum contains spikes at the correct amplitudes at only positive frequencies, which are the only ones with physicality."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}