dft-solutions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[exercises](dft.ipynb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "amplitudes = 1, 0.75, 0.5, 0.25, 0.125\n",
    "frequencies = 200, 400, 600, 800, 1000  # Hz\n",
    "phaseshifts = 0, 0, 90, 90, -90  # degree\n",
    "duration = 1  # second\n",
    "fs = 44100  # Hz\n",
    "\n",
    "# parameters as row vectors\n",
    "amplitudes = np.asarray(amplitudes).reshape(1, -1)\n",
    "frequencies = np.asarray(frequencies).reshape(1, -1)\n",
    "phaseshifts = np.asarray(phaseshifts).reshape(1, -1)\n",
    "\n",
    "# time values as column vector\n",
    "t = (np.arange(np.ceil(duration * fs)) / fs).reshape(-1, 1)\n",
    "\n",
    "cosines = amplitudes * np.cos(2 * np.pi * frequencies * t + np.deg2rad(phaseshifts))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Split into several variables (e.g. for listening), each holding a row (no copies of the data are made!):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x1, x2, x3, x4, x5 = cosines.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x6 = np.sum(cosines, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "six_signals = np.column_stack((cosines, x6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Kzp0H1ttGEMTh8TvviP73jz8W/e7WIJMBzzwDrF49BUAgFi8GSksbL7dE20OnK4cgKNGh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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7efe0023a550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(six_signals[:500])\n",
    "plt.ylim(-2.3, 2.3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When playing back with a sound card or when writing to a sound file, the values are automatically clipped to a range from -1 to 1, leading to heavy distortions for signals that exceed this range:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ySUReF5Fvph5P3P4QkVYiskpE1qX2xazU44nbF2ki0kxE1orI/NTPidwXIlIu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\nIU9E5LH/B/9+9bjPIofKAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7efdfdff1c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.clip(x6[:500], -1, 1))\n",
    "plt.ylim(-2.3, 2.3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To avoid this clipping, we should scale the signal values to $\\pm 1$ beforehand:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x6_norm = x6 / np.max(np.abs(x6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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s1GUNQjkQEZFnMmLtGiIiCh+7a4iIPMYgT0TkscCDvIi0EZFFIvKFiNwe9PGi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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7efdfdf30630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x6_norm[:500])\n",
    "plt.ylim(-2.3, 2.3);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def naive_fourier_transform(x):\n",
    "    \"\"\"Naive implementation of the DFT equation.\n",
    "    \n",
    "    The input must be a one-dimensional array.\n",
    "    WARNING: this implementation is extremely inefficient!\n",
    "    \n",
    "    \"\"\"\n",
    "    N = len(x)\n",
    "    n = np.arange(N)\n",
    "    return np.array([np.sum(x * np.exp(-1j * 2 * np.pi * k * n / N))\n",
    "                     for k in range(N)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In case you wonder what that last thing is, up there inside of the `np.array(...)`: it's a *list comprehension*!\n",
    "This is a really nice feature of Python, to learn more about them, have a look at [List Comprehensions, Generators, Generator Expressions](http://nbviewer.ipython.org/github/mgeier/python-audio/blob/master/misc/comprehensions-and-generators.ipynb)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#X1_naive = naive_fourier_transform(x1)  # takes much too long!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Complexity of naive implementation: $\\mathcal{O}(N^2)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X1_naive = naive_fourier_transform(x1[:1000])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Matrix multiplication:\n",
    "One could calculate the DFT by multiplying a matrix containing the values of the exponential function by the vector $x$.\n",
    "The computation could *theoretically* be a bit more efficient, since the whole matrix multiplication can be done in C or Fortran (or whatever NumPy uses) instead of native Python.\n",
    "But this only matters for very short signals, because the memory requirements would be quadratic instead of linear!\n",
    "For example, storing the above-mentioned matrix for a one-second signal @ 44.1 kHz would need ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "15558480000"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "44100**2 * np.dtype('float64').itemsize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... thats about 14.5 GB of memory!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X1_fft = np.fft.fft(x1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Algorithmic complexity of FFT: $\\mathcal{O}(N\\log_2N)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X1_fft = np.fft.fft(x1[:1000])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.8615763522521785e-11"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.max(np.abs(X1_fft - X1_naive))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "356.59263628672301"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.max(np.abs(X1_naive))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Maximum error in decibel (where 0 dB is the absolute maximum value of `X1_naive`):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-253.13325467955931"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "20 * np.log10(np.max(np.abs(X1_fft - X1_naive)) / np.max(np.abs(X1_naive)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Maximum dynamic range of a 16-bit signal: about 96 dB.\n",
    "This does not depend on the sampling rate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "96.329598612473987"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "20 * np.log10(2**16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "cosine_spectra = np.fft.fft(cosines, axis=0)\n",
    "X6 = np.fft.fft(x6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Using matplotlib backend: TkAgg\n"
     ]
    }
   ],
   "source": [
    "%matplotlib\n",
    "plt.plot(np.abs(cosine_spectra))\n",
    "plt.figure()\n",
    "plt.plot(np.abs(X6));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note the warning message when plotting a complex signal!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# TODO: phase plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's probably best to write your own plotting functions for spectra, but you should also have a look at [matplotlib.pyplot.magnitude_spectrum()](http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.magnitude_spectrum) and [matplotlib.pyplot.phase_spectrum()](http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.phase_spectrum)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x1_ifft = np.fft.ifft(cosine_spectra[:, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.plot(x1_ifft.imag)\n",
    "plt.figure()\n",
    "plt.plot(x1_ifft.real - x1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The maximum of the imaginary part is in the order of magnitude of $10^{-15}$, same for the difference of the real part compared to the original signal $x_1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# TODO: plot function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# TODO: plot function using rfft()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which parameters are visible?\n",
    "\n",
    "* time domain: amplitude, frequency, duration. Also the phase, if you look closely.\n",
    "\n",
    "* frequency domain: amplitude, frequency. Phase not that easily. Duration not visible."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
    "  <a rel=\"license\"\n",
    "     href=\"http://creativecommons.org/publicdomain/zero/1.0/\">\n",
    "    <img src=\"http://i.creativecommons.org/p/zero/1.0/88x31.png\" style=\"border-style: none;\" alt=\"CC0\" />\n",
    "  </a>\n",
    "  <br />\n",
    "  To the extent possible under law,\n",
    "  <span rel=\"dct:publisher\" resource=\"[_:publisher]\">the person who associated CC0</span>\n",
    "  with this work has waived all copyright and related or neighboring\n",
    "  rights to this work.\n",
    "</p>"
   ]
  }
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