{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*This post was entirely written using the IPython notebook. Its content is BSD-licensed. You can see a static view or download this notebook with the help of nbviewer at [20160907_ImpulseResponses.ipynb](http://nbviewer.ipython.org/urls/raw.github.com/flothesof/posts/master/20160907_ImpulseResponses.ipynb).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we're going to explore the fascinating world of impulse responses. This blog post is inspired by many interesting projects of Allen Downey and in particular [this student project of his](https://t.co/XV1a3o76Q7). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# What's an impulse response?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In many fields, the output of a system can be represented as the transformation of an input by a function, which we will call G. Symbollically, we would write this:\n",
    "\n",
    "$$\n",
    "y = G * x\n",
    "$$\n",
    "\n",
    "It turns out that if you choose G wisely, you can solve many problems of physics. The name G is chosen after George Green, who invented the concept. For a good introduction about Green's functions, I recommend this [Wolfram Blog entry](http://blog.wolfram.com/2016/03/31/new-in-the-wolfram-language-greenfunction-and-applications-in-electricity-odes-and-pdes/).\n",
    "\n",
    "Green functions come in many variants, depending on the problem to be solved, see [Wikipedia](https://en.wikipedia.org/wiki/Green%27s_function). In this posting, I propose to do the followine:\n",
    "\n",
    "- demonstrate the use of an impulse response on an audio file for different environments\n",
    "- transform the chunk of a song to something that gives the listener an impression to move in space\n",
    "\n",
    "First, let's introduce the theory background we'll need to perform our task."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Theory "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Actually, we need to introduce two theory items in this notebook: the convolution theorem, which is a classic signal processing way of teaching and the wave equation solution using a Green's function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The convolution theorem "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The convolution theorem states that to apply a certain function G, I can use the Fourier frequency space to do it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving the wave equation... in two dimensions! "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All sounds that move through space obey the wave equation. It turns out that the wave equation can be solved with the following Green's function: \n",
    "\n",
    "$$\n",
    "G(t, \\rho) = {\\displaystyle {\\frac {1}{2\\pi c{\\sqrt {c^{2}t^{2}-\\rho ^{2}}}}}\\Theta (t-\\rho /c)}\n",
    "$$\n",
    "\n",
    "Here $\\rho$ is the distance to the receiving point, $c$ is the celerity of the medium and ${\\displaystyle \\Theta (t)}$ is the Heaviside step function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Applying an impulse response to a sound"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As explained above, let's now apply an impulse response to a sound. We will use a sound from FreeSound to test our example. First, we need to download our sound as a WAV file to work with it. To do this, we write a function that downloads it from the internet, stores it in a temporary file and uses FFMpeg to convert to WAV:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import urllib\n",
    "import tempfile\n",
    "\n",
    "def download_mp3_as_np_array(url):\n",
    "    \"Downloads a mp3 file and converts it to a numpy array.\"\n",
    "    pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now download the sound segment that we will use in this post:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "snd = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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