python-data-project-scipy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Signal processing using Scipy\n",
    "\n",
    "In this notebook we will explore how the [scipy](https://scipy.org/scipylib/) package can be used to perform signal processing.\n",
    "\n",
    "Scipy stands for *sci*entific *py*thon, and is a package with a vast range of modules for a range of scientific computing problems. From linear algebra to optimization; from integration to interpolation: it is the go-to Python package for traditional scientific computing problems. Scipy is built from numpy's `ndarray`, and scipy interfaces with `matlotlib` for visuals. So when working with scipy it is essential we also have numpy and matplotlib imported too."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scipy is a large package with many submodules. It is standard to just import the modules we will be working with on a givgen task. In this notebook, we'll be using the modules `fftpack` and `signal`, which give access to methods related to the Fast Fourier Transform, and signal processing, respectively. We will also need some helper functions for reading in signals, from the `scipy.io` module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import fftpack\n",
    "from scipy import signal\n",
    "from scipy.io import wavfile"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Working with signals\n",
    "A signal is a series of measurements over time, related to some phenmomenon. It could be the light emitted from a chemical reaction, sound transmitted through the air, or the heat of an organism throughout some biological process. When we work with signals in python, we are considering digital signals: samples of a signal taken at discrete time-steps. Today we will be working with a familiar signal: we'll be working with audio signals (i.e. those coming from sound waves). \n",
    "\n",
    "### Reading in a wav aoudio file.\n",
    "Let's see how we can represent an audio signal in python. We are going to use the `wavfile.read` helper to read in a recording of a guitar chord (you can listen to it by clicking on the file in the Jupyter folder)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.6/site-packages/scipy/io/wavfile.py:273: WavFileWarning: Chunk (non-data) not understood, skipping it.\n",
      "  WavFileWarning)\n"
     ]
    }
   ],
   "source": [
    "chord = wavfile.read(\"data/chord-11.wav\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function returns a tuple. The first element is the sample rate, and the second is the actual data from the file."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(44100, array([[ -8,  45],\n",
       "        [ 30,  91],\n",
       "        [ 82, 129],\n",
       "        ...,\n",
       "        [  0,   0],\n",
       "        [  0,   0],\n",
       "        [  0,   0]], dtype=int16))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chord"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting the data from the wav with `plt` will give the charactersitic plot of the sound wave that we're used to."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fc1db38bda0>,\n",
       " <matplotlib.lines.Line2D at 0x7fc1db397a58>]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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8zuvuLHKIlDxEuqLZl1J110So+BAa9gYdjSQgJQ+RLip37yb43Wg2Pvplr0t0ogyrL3FByUOkixuwfi785jg2Pn0T1GyBJO1EI5Gl5CEiABQs+yPceRRrnrnDe/GUkoi0QclDRADIpA6Aov/cAXePY9NL98C615VEpFVKHh2xRy+Akq4jzX+fep9Xb4Y/nsvG1/7s3VgXCaMnzDug9ucjyAo6CJEYC9EEQPo/fwAvbmXzBX+j34BB0Ks42MCiqbEBGvZ4Q7o07IWmBqivBUvxytN7esPs9CgMOtLAKXl0QFZjTdAhiASmsNF7Y0L63KugsZyqL71Gbu8i6JYXcGSHobEBKlbR1KMvtav+RWNaD+peu4u0jExy1zxHRf8zKNj0CrVp+WTVHzgGa1XecTRkFZKy/UN2FRxHxhElpPcfRcbSxwgd/3nSew2Bnn1h2yov2aakeWO5JdHwMBqepCM0NInIfuU9j6GwZiWN311HKCs36HA6prGe6lfvJTT4FFL+fBHbis5m0NonqUjrR0F9dB6WLMs6lqLaFWzNOY7e1e+y9tiZDF5xH5UX/JWm5c+SP/kmal+7mx6Tb4O9NdCjd/sbjYGODk+i5NERSh4iB1V1xu3kfOIK6jYsIWNQCdRsgvyhwQW0YzN0L2Tjs7eTP/Icuv1pMhv7jGfAln8GF1MbKnoUU7BzFWXFl1O06k/s+Mp/yO47LLB4lDyUPERi7qPs4+i74102TvoDje8+Tf8pt9G4ZQUZJ3yu/ZU7a88OSOvGqgUP0GvEeHb+37dIG/MF+j7/1f1xJKIlp93LiWddEtj+lTyUPETixsYhU8nd+Ap15/+O+p0V9B7+Se8tkwWd+B/2znJITWfvncdRM2QyvT6YzfqC0xlU8Wr0Ag/AqpxTKa5+nT3Xf0hmdq+Y71/JQ8lDJO7tyB+FS82krK47dak96Nn/GCwzh24pDaTVbCC1ez628hnIKiD3o9fZY93IdLuDDjsmPpjyHEedeFrM96uXQYlI3MuufA+A/f8923aQF4Lt8L66SuIAMIvv/9jrIUERkThkKHmIiEgnxfsTIUoeLTTWVlHx2sNBhyEiXVzINcb10EhKHi2UPjiDgheuY9PKN4MORUS6sLQXvgc/HUT93tqgQ2mVkkcLTf6rOau3bw84EhHpygbsWg7Ajh3x2fpQ8mihyT8kTXqrmojEgaaG+qBDaJWSRwvOQgA0NTaydslLbLp9RMARiUhX1tQYn8lDz3m0kNbk9SMf9dK0gCMREYHtmz5k66r/MGpCFIZ4OQxKHi30qv8o/vvIiUiXccw8L2m4My7CUuLnYlH8RBIn1maPDjoEEZED1NXtCTqEZpQ8WjA1O0QkDtXtVfI4JGY22czeN7NSM7sxWvtxSfSmLxFJHmm/Ppamhoagw9gvIZKHmYWAu4BzgeHAJWY2PEp7i85mRUQOQ6bbQ8rtBUGHsV9CJA9gLFDqnFvtnKsDZgNTorGjFBc/mV1E5ACzclj74UrKKyoCDSNRelsNADaEzZcBJ0djR/WZBVATjS2LiETG4D81//O3MGdys/lek65nyIio/IncL1GSR2vXkg4Yr9jMZgIzAQYNGnRIO1qfO5ax5U8f0roiIkEYUL242R/Jyurot0oSJXmUAQPD5ouATS0rOefuA+4D702Ch7KjqZd9BWbdcCiriojE3LJekxlx7ePNyvrFYL+JkjwWAsVmNgTYCFwMXBpsSCIisbN43O8ZM/nyA8qDGkApIZKHc67BzK4F5gMh4CHn3LKAwxIRiZnWEkeQEiJ5ADjn5gHzgo5DREQSp6uuiIjEESUPEZE49kHaMXBrZdBhHEDJowNq6B50CCLSRdX2OxlSQkGHcQAlj1aUjr+72fyyvLMCikREurrjp98ZdAitUvJoxZHjL+Pt4uv3z6fv3RZgNCLSlVkoLegQWqXkcRD5R54EwDt556DBEkUkCItH3hJ0CAel5HEQR449l1UnfZ9jrrhXw7SLSMx9EDqSMVO/E3QYB5Uwz3nEnBnFn/rGvplAQxGRrsfF+d8dtTw6Qi0PEYkxZ/HXwyqckkcHxPv/AEQk+TQccXrQIbRJyaNDlDxEJHZq8kcy8rKfBR1Gm5Q8OkTJQ0RiYxOF9Pz6v+PywcBwSh4dodwhIlFWESrEYdiZ/xN0KB2i3lYdYAe+tFBE5JBUZA6isqCElCNOJrd7Jt2OKGHnR6X0HDoWy+8fkxc5RYKSRwekKHmIxFR5n9PJm3At6/77CkUTZ/LRgt+QdcKF7H7pF/Q4/tNkvfZTdoz+Kpnv/on6PieQu3Yeoaa6oMM+QHm/CYTqa3Cnf4uU9O7kDSuhICWVgtSMZvWyikYGFOGhM+eS8w9jSUmJW7RoUUS29d87P8PxNa9EZFsiArWhbLIad7Dt1FsJvfc4uVc9i1WUwhGnHvpGq9ZD/W4IpYFzVGyvJCWtGzWbS8nM70fjP3+BKz6brCUPUH/85RS+egvrB05h0IY5VGf0I2fvZspyxlBUvZjynsMprFlOTbcieu4uY0e3gWTv3sCWgpPpU/EWG3LHMrDqbbaM+DJ9lj3A1uFfpPfyh6mZ/jJ12zdQMHQ01NdC4dGRO2gxYmaLnXMl7dZT8mjfqz+cxOmNb0VkWyJd0caexzOg5r/s+uwf2b18PgUX3oFVrob+JwYdGpS+CAPHQuVq6DOSps3vktLveNj8DvQ9DlbNh+JJULYQisbCjjLo2R92boHcgUFHH3FKHhFMHit/eS7H7Hg9ItsS6QrKck+iz84V1J3/e3bvrSf/hPNJycgKOizpgI4mD93z6IC+n7oJHvt00GGIxL0do68m++QvUJQ3GNK7kwZ6G06SUvLogJTCY4IOQSTu7M4awJ4+o0kfdwWpuQPJ6FNMdtBBScwoeXREih70EKlL7Ul6Qw1Vn5+LpWWQc+Q4ugUdlARGyaMDTE8JShfUhJGCo/rMn9NUtYG8yTcDkJuWGXBkEg+UPDrAUvQgvnQd5UddQuEHj8F179FYs5mcQWODDknikP4qdoBaHtIVVB81FYDCS+6BWdWk5A0kpMQhB6GWRweY7nlIEmokhRBNVE6+C7f+bQqm/hpSHgw6LEkQSh4dYGqgSRLaMuFO+vftR/7R58K4LwQdjiQYJY8OUMtDksmWMddTkN2d/qdPj/thvyV+KXl0iJKHJK4GUkmlgV2fvJXubhd9zrgBWgzMJ9JZuh7TAe21PEpP+en+6cXDb4p2OCKdUjvmKrjyJbpPuB7OvFWJQyJCyaMDUuzgh6mGLOqzBwOwIm0EodS0GEUl0VCd7Y2CuvP4KwKO5PCVD7sIvjiP7PO+DwPGgKkFLZETteRhZrPMbKOZveN/zgtbdpOZlZrZ+2Z2Tlj5ZL+s1MxuDCsfYmZvmdkqM3vczNKjFfdBfktbS8OeQHeMOnliLEKSCNuQPpSmopPJmf4YzKqmxwW/omrs9VSdlhhvdQu3Z8jZcOVLFF58Nwz+hDdEuUiERfuex6+cc78ILzCz4cDFwAigP/CCmR3lL74LOBsoAxaa2Vzn3HLgZ/62ZpvZvcAM4J4oxx4Wcxs5duSF7LsnYkBowIksGX4DJy6P75fXi6ey+zDyd31I/vhrSTl1RrNluefdBkB992zo2Ye0p6YFEWKH7Bx+MamjLiAzr4jM/CGQruEIJbqCuGE+BZjtnNsLrDGzUmDfk0ilzrnVAGY2G5hiZiuAicClfp1HgFnENHk0b3lsPuIz9Fs3l2p6knPBr0n7z0sAhPwn0evTesYqNDlEdZZB+ufuJ3/w6ZCV3+bIr2mnzPQmuj8D2QPgd6OpHfhJsjb8KyaxHsye7KFknvcjKBhGjwR86ZAktmgnj2vNbBqwCPiWc247MAB4M6xOmV8GsKFF+clAAVDlnGtopX4zZjYTmAkwaNCgSP2GA5JH97O/Bw/MZU+PgeSEUhk2rBiAAWPOa211iTP1xecSOvkqOHJC51Yc8knv+8b1ZKV2g3cfh0Hj2POXy7CJN9P41gNknPwlQk9/iabP/J6Uze/A2Kto+L+rqck+mrwVfz7s2KtP+gaWlU/2qTPITEkFjTMlATms5GFmLwB9W1l0M17L4IeA87/vBK6g9X6vjtbvv7g26h9Y6Nx9wH3gvQyqnfA7rOVlq+wBx1A34VZ6n/B5b3n+EPjmcrJ6Jsqr67uu2pJryDr/x4e3kcwc73v05d7sdW9786OmeN8jppCSEgK85alXvkBeQx38dyyNmfmEnvwCu1Nz6dZQdcCmd2f2ptuerdQOPIOsDa+w98KHcbvKyTzRO9dy9u1bJGCHlTycc2d1pJ6Z3Q8868+WAeHvbiwCNvnTrZVvA3LNLNVvfYTXj42wlkeDSyHVjPQzvtW8Ts7HjaGBhXmxikw6qKb4Anpe9jAxeZddaw/epabDmOmEAEZUk1H6T5rmXE3TZ++hfuHDhAqLSX/1Z3T72htQs5msvqMAUKdaiVdRu2xlZv2cc5v92QuApf70XOCvZvZLvBvmxcDbeC2MYjMbAmzEu6l+qXPOmdnLwFRgNjAdmBOtuFtlRt0p15H+xm9oSmm/50q/Uy+BF66JQWDSIbOqibe7UClHjodvLScFSB12hld45ve87+69ggpLpMOiec/jDjM7Ae8S01rgKgDn3DIzewJYDjQA1zjnGgHM7FpgPhACHnLOLfO3dQMw28xuB5YAMR+9LX3S96nbvo7QSTPar5wS4j89zmD0zleiH5gc3EUPQk5R0FGIJKWoJQ/n3OVtLPsR8KNWyucB81opX83HPbKCYUb6xY8EGoJ00LQ5kN4DikqCjkQkaWlsK0k+Q8cHHYFI0lPykITXkJ5Dal01fOdDcBHrZCcibVDykISX+u0V0NQImdlBhyLSZWhgxBjZMOxSmkJ6oCsS6kZ6zzxw7WL44jxvKA4lDpGYUvKIEmvxHGPV0E+T8j9bAoomsW072huZZu8nva6s6VPvg1nV0OtIb+A/EYk5XbaKlhbX3nUpvvMa0nqSWl9DwTnfhUvu8R6Ym3hD0GGJCEoeUWM0tShpOS8Hs6b3WQzZ+gKp17wONR95w7+ISFxR8oiSlpetrKnhIDVln0ZChGhkwOmXw6invcLcyA1wKSKRo+QRLS2uUw3OaWW8IwGgiRRSaKLxnJ8RKhxK+jC9UEsk3il5RMm+y1YrjruRYxtW0HPEpIAjil9rep/FsHO/Tvrg0/SqVJEEod5WUWJ+y6M2ewj8v0chVeOjtlSX7o0+PPTMK2DI6UocIglELY+o8ZJHSluvsO2itpFLr6NPJX3CzdB3ZKsvbBGR+KbkES3Ou2xlKUoeLWX3zIZLHgs6DBE5DEoeUbK/t5VaHs2d9k3ST7gs6ChE5DApeUSJ+S2PlNbeKtdF1ZdcSdpZs4IOQ0QiQMkjWvZdttJNYM+satp/B6OIJAoljyjZd9mqI/c8GkkhlKxPoI+cCsecF3QUIhJhuiAfJQXdvf9n983JarviN5ayPW9UDCIKyNQHYeRFQUchIhGm5BElvSd/F4BeQ09ovcJX/g1XzIfcgfSa+HUAdtJOokkw7sizgw5BRKJEl62ixI4+1xs2/GD6jvx4etRUGDWV6u8fRQ9XG/3gYmFWtZ7fEEliSh5xpOVgiglp2ETIHxZ0FCISZUoeElmX/2/QEYhIDCh5xNqMBbB1RauLTG+MEpEEoeQRawPHep9WZKalQH2M4zlMrkcfbOcWuGFd0KGISAwpecSRnGFjYeUzQYfRKfaN97wHItO6BR2KiMSQuurGkZQL/wAzXqCxW6+gQ2nfxFug+BxvqHklDpEuRy2PeJLeHQaeRGjGfPjg7/D8LUFHdKCvvAYVpTDigqAjEZEAqeURj3odCad+LegoWtd3lBKHiCh5SMfUT/oJ9Ds+6DBEJE4cVvIws8+Z2TIzazKzkhbLbjKzUjN738zOCSuf7JeVmtmNYeVDzOwtM1tlZo+bWbpfnuHPl/rLBx9OzAnlmrfhgj8EHQUATWOugKv+FXQYIhInDrflsRS4EGj2V8XMhgMXAyOAycDdZhYysxBwF3AuMBy4xK8L8DPgV865YmA7MMMvnwFsd84dCfzKr9c1FB4Nx1+MO+b8oCNBj6CISLjDSh7OuRXOufdbWTQFmO2c2+ucWwPjlTGYAAAHwklEQVSUAmP9T6lzbrVzrg6YDUwx76UXE4Gn/PUfAT4btq1H/OmngDOti70kwy7+S9vjZMVAZppeaiUiH4vWPY8BwIaw+TK/7GDlBUCVc66hRXmzbfnLq/36BzCzmWa2yMwWlZeXR+indG3bT70ZCo+FFHXME5GPtZs8zOwFM1vaymdKW6u1UuYOobytbR1Y6Nx9zrkS51xJYWFhG+ElqBkvwKVPxnSXoeMugmvehK7V2BORdrT730nn3FmHsN0yYGDYfBGwyZ9urXwbkGtmqX7rIrz+vm2VmVkqkANUHkJMiW/gSd73V16D+t3wYPTel7E3qy8ZtR+RnZ0XtX2ISOKK1mWrucDFfk+pIUAx8DawECj2e1al491Un+ucc8DLwFR//enAnLBtTfenpwIv+fW7rr6jDjo+VqR8eMpP4Oo3ISs/qvsRkcR0uF11LzCzMuAU4Dkzmw/gnFsGPAEsB/4BXOOca/RbFdcC84EVwBN+XYAbgOvNrBTvnsaDfvmDQIFffj2wv3uvRM+g3vnQ+9igwxCROGXJ+p/4kpISt2jRoqDDiK5ZORHfZEX+iRRULoFvLoOcoohvX0Tim5ktds6VtFdPT5hLM2uGXALf26zEISJtUvJIZBP/B8bfFNFNDunVA9KzIrpNEUk+Sh6J7JPfhvGRuQVU0d1773jBsNER2Z6IJDclDwFgR/EF8L1NukkuIh2i5CEADOlT4L1PRESkAzTmRDI47xeQVQBPfanTqy7OO48xo0bCSTParywi4lPLIxmMvRJGXnhIqxbk5cDEm73XyYqIdJBaHslkwi1QeBQ01EHfkXD3uHZXGTxseLt1RERaUvJIJmd854Ail5GNnXED1Rl9yXlmBjvS+5DdWAXT5kD9Lhg6MYBARSTRKXkks6kPYf1HQ/4Qcj5aCkDPwkFw5QcBByYiiU7JI5mNvOjj6T4j4IwbsdGXBxePiCQNJY+uwgwmRPZpdBHputTbSkREOk3JQ0REOk3JQ0REOk3JQ0REOk3JQ0REOk3JQ0REOk3JQ0REOk3JQ0REOs2cc0HHEBVmVg6sO8TVewHbIhhOMtIxapuOT/t0jNoXxDE6wjlX2F6lpE0eh8PMFjnnSoKOI57pGLVNx6d9Okbti+djpMtWIiLSaUoeIiLSaUoerbsv6AASgI5R23R82qdj1L64PUa65yEiIp2mloeIiHSakkcLZjbZzN43s1IzuzHoeKLJzAaa2ctmtsLMlpnZdX55vpktMLNV/neeX25m9lv/2LxrZqPDtjXdr7/KzKaHlY8xs/f8dX5rZhb7X3p4zCxkZkvM7Fl/foiZveX/1sfNLN0vz/DnS/3lg8O2cZNf/r6ZnRNWnvDnm5nlmtlTZrbSP5dO0TnUnJl90/83ttTMHjOzzIQ/j5xz+vgfIAR8CAwF0oH/AsODjiuKv7cfMNqf7gl8AAwH7gBu9MtvBH7mT58H/B0wYBzwll+eD6z2v/P86Tx/2dvAKf46fwfODfp3H8Jxuh74K/CsP/8EcLE/fS/wVX/6auBef/pi4HF/erh/LmUAQ/xzLJQs5xvwCPBlfzodyNU51Oz4DADWAN3Czp8vJvp5pJZHc2OBUufcaudcHTAbmBJwTFHjnNvsnPuPP10DrMA70afg/UHA//6sPz0FeNR53gRyzawfcA6wwDlX6ZzbDiwAJvvLsp1zbzjv7H80bFsJwcyKgE8BD/jzBkwEnvKrtDw++47bU8CZfv0pwGzn3F7n3BqgFO9cS/jzzcyygU8CDwI45+qcc1XoHGopFehmZqlAFrCZBD+PlDyaGwBsCJsv88uSnt80PhF4C+jjnNsMXoIBevvVDnZ82iova6U8kfwa+C7Q5M8XAFXOuQZ/Pvw37T8O/vJqv35nj1siGQqUA3/0L+09YGbd0Tm0n3NuI/ALYD1e0qgGFpPg55GSR3OtXUtN+u5oZtYDeBr4hnNuR1tVWylzh1CeEMzsfGCrc25xeHErVV07y5Ly+PhSgdHAPc65E4FdeJepDqbLHSP/fs8UvEtN/YHuwLmtVE2o80jJo7kyYGDYfBGwKaBYYsLM0vASx1+cc3/zi7f4lwvwv7f65Qc7Pm2VF7VSnig+AXzGzNbiXQqYiNcSyfUvP0Dz37T/OPjLc4BKOn/cEkkZUOace8uffwovmegc+thZwBrnXLlzrh74G3AqCX4eKXk0txAo9ntBpOPdrJobcExR419HfRBY4Zz7ZdiiucC+3i7TgTlh5dP8HjPjgGr/ksR8YJKZ5fn/y5oEzPeX1ZjZOH9f08K2Ffecczc554qcc4PxzoWXnHOXAS8DU/1qLY/PvuM21a/v/PKL/V40Q4BivJvACX++Oec+AjaY2dF+0ZnAcnQOhVsPjDOzLP837DtGiX0eBd0TId4+eL1BPsDrvXBz0PFE+beehte8fRd4x/+ch3d99UVglf+d79c34C7/2LwHlIRt6wq8G3ilwJfCykuApf46v8d/MDXRPsB4Pu5tNRTvH20p8CSQ4Zdn+vOl/vKhYevf7B+D9wnrLZQM5xtwArDIP4/+D6+3lM6h5sfo+8BK/3f8Ca/HVEKfR3rCXEREOk2XrUREpNOUPEREpNOUPEREpNOUPEREpNOUPEREpNOUPEREpNOUPEREpNOUPEREpNP+P49AHBvKxR/qAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(chord[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To check we're doing thing correctly, let's write that data back to another file. By opening the original file and the file we've written, you should convinve yourself that the numpy array created by `wavfile.read` really does represent the audio signal of the wav file."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "wavfile.write(\"data/test.wav\", chord[0], chord[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Applying the FFT\n",
    "The Fourier transfrom is a way of breaking down a signal into its component parts. With audio signals, it tells us how we could have built up our original signal by combining certain elementary (sin and cosine) sound waves. It's a bit like \"unmixing\" coloured paint to find out how much red, blue and yellow went into a given colour. The process is quite complicated, so for the purposes of this notebook we will just understand it at this high level: given a signal, the Fourier transform gives us its decomposition into elementary waves, which is like a kind of fingerprint unique to that signal. The Fast Fourier Transform (FFT) is a particularly efficient algorithm for calculating the Fourier transform of a signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.6/site-packages/numpy/core/numeric.py:492: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  return array(a, dtype, copy=False, order=order)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fc1db2e8a20>,\n",
       " <matplotlib.lines.Line2D at 0x7fc1db2e8b70>]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "chord_transform = fftpack.fft(chord[1])\n",
    "plt.plot(chord_transform)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 37.+0.j, -53.+0.j],\n",
       "       [121.+0.j, -61.+0.j],\n",
       "       [211.+0.j, -47.+0.j],\n",
       "       ...,\n",
       "       [  0.+0.j,   0.+0.j],\n",
       "       [  0.+0.j,   0.+0.j],\n",
       "       [  0.+0.j,   0.+0.j]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chord_transform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}