Lessons/FourierAnalysis/2_NumericalFFTs.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical Fourier Transforms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "%pylab inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this exercise we'll explore how one can take Fourier transforms numerically using Python. Recall that the Discrete Fourier Transform (DFT) of a discrete set of points $g_1 = g(t_1)$, $g_2 = g(t_2) $, ..., $g_N = g(t_N = T)$ is defined to be\n",
    "\n",
    "$\\tilde{g}_n \\equiv \\sum_{m=1}^N g_m \\exp \\left( \\frac{-i 2 \\pi m n }{N} \\right)$\n",
    "\n",
    "which tells us how much of each complex exponential is needed to reconstruct the series of $g_m$ values. This reconstruction is accomplished by the inverse Discrete Fourier transform, given by\n",
    "\n",
    "$g_m \\equiv \\frac{1}{N} \\sum_{n=1}^N \\tilde{g}_n \\exp \\left( \\frac{+i 2 \\pi m n }{N} \\right)$.\n",
    "\n",
    "Python has built-in functions to compute Fourier transforms. They can be found under `np.fft`, where \"FFT\" stands for \"Fast Fourier Transform\". The FFT is mathematically the same as the DFT, but uses various internal computing tricks to make things \"fast\". Let's try it out!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tophat Example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The tophat function is a function is a function that is $1$ close to the origin but zero everywhere else."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def tophat(t,a):\n",
    "    out = t.copy()\n",
    "    for i,elem in enumerate(t):\n",
    "        if abs(elem) > a:\n",
    "            out[i] = 0.\n",
    "        else:\n",
    "            out[i] = 1.\n",
    "    return out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m_coords = np.arange(-1024, 1024)\n",
    "gt = tophat(m_coords,150)\n",
    "\n",
    "f, ax = plt.subplots(figsize=(9,6))\n",
    "ax.plot(m_coords,gt,lw=2,c='red')\n",
    "ax.set_xlabel(r\"$m$\")\n",
    "ax.set_ylabel(r\"$g_m$\")\n",
    "ax.set_ylim(-0.2, 1.2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, you'll see that rather than having $m$ start at zero, I decided to have it be centered around zero, so that the $g_m$'s start from $m= -N/2$ rather than $m=0$. This is typical of a lot of our data. However, the slightly irritating thing about most numerical FFT routines is that they assume we're starting at $m=0$ Luckily, `numpy` provides a handy routine called \"fftshift\" that can be used to put our data into the right format."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 648x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(figsize=(9,6))\n",
    "ax.plot(np.fft.fftshift(gt),lw=2,c='red')\n",
    "ax.set_xlim(0,2048)\n",
    "ax.set_ylim(-0.2,1.2)\n",
    "ax.set_xlabel(\"Array index\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed, the first element of the array now corresponds to $g_0$. But notice how the part of the function immediately to the left of $m=0$ is now on the far right! It turns out that FFTs assume that axes \"wrap around\". So in our new ordering scheme, the data goes from $m=0$ to $m=-N/2$, and then goes from $t=-N/2$ back up to $t=0$. Thus, the left half of the bump at appears on the far right. Said differently, FFTs assume that our function is periodic. In this way, FFTs are actually quite closely related to Fourier *Series* (which, remember, are for like Fourier transforms for periodic functions)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The FFT algorithm also orders the *output* of the FFT with $\\tilde{g}_0$ as the first element. Thus, the output needs to be adjusted too, and the whole process is given by an initial np.fft.fftshift, the actual FFT using np.fft.fft, and the reordering of the final result using another np.fft.fftshift:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "gtilde = np.fft.fftshift(np.fft.fft(np.fft.fftshift(gt)))\n",
    "\n",
    "f, ax = plt.subplots(figsize=(9,6))\n",
    "ax.plot(gtilde.real,lw=2,c='red')\n",
    "ax.plot(gtilde.imag,lw=2,c='blue')\n",
    "ax.set_xlim(512,1536)\n",
    "ax.set_xlabel(\"n\")\n",
    "ax.set_ylabel(r\"$\\widetilde{g}_n$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is known as a <b> sinc </b> function.  What happens because of the symmetry of the original function? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks like the function $\\tilde{g} (f) = \\sin (2 \\pi f) / \\pi f$ that we would've predicted by doing the analytic version of the Fourier transform. But whereas the pencil-and-paper version gave us a continuous function of $f$, here we have a bunch of values in an array, and we need to know which value of $f$ each element of the output array corresonds to. It turns out that numpy has a function called np.fft.fftfreq for this. As input, it needs to know the length of the original input array to the FFT, as well as the spacing in $f$ that is assumed there."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The effect of width"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at what happens when we change the *width* of the tophat function.  Here we're going to have to take seriously how to get the x-axis of the Fourier transform labeled properly so we can interpret it. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gt(t,a):\n",
    "    out = t.copy()\n",
    "    for i,elem in enumerate(t):\n",
    "        if abs(elem) > a:\n",
    "            out[i] = 0.\n",
    "        else:\n",
    "            out[i] = 1.\n",
    "    return out\n",
    "\n",
    "def gtilde(f,a):\n",
    "    return a* np.sinc(f * a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.02\n"
     ]
    }
   ],
   "source": [
    "t, dt = np.linspace(-10.,10.,1000,retstep=True,endpoint=False)\n",
    "print(dt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(figsize=(9,6))\n",
    "ax.plot(t,gt(t,1.),lw=2,c='blue',label='Narrow')\n",
    "ax.plot(t,gt(t,2.5),lw=2,c='red',label='Medium')\n",
    "ax.plot(t,gt(t,5.),lw=2,c='black',label='Wide')\n",
    "ax.axhline(0, color='black',ls='--')\n",
    "ax.axvline(0, color='black',ls='--')\n",
    "ax.set_ylim(-0.2, 1.2)\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Make the frequencies corresponding to the time sampling\n",
    "freq = np.fft.fftfreq(n=len(t),d=dt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(figsize=(9,6))\n",
    "ax.plot(freq,gtilde(freq,1.),lw=2,c='blue',label='Narrow')\n",
    "ax.plot(freq,gtilde(freq,2.5),lw=2,c='red',label='Medium')\n",
    "ax.plot(freq,gtilde(freq,5.),lw=2,c='black',label='Wide')\n",
    "ax.axhline(0, color='black',ls='--')\n",
    "ax.axvline(0, color='black',ls='--')\n",
    "ax.set_xlim(-5,5)\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Fourier transform of a Gaussian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Gaussian function is given by\n",
    "\n",
    "$g(m) = \\frac{1}{\\sqrt{2 \\pi a^2}}\\exp \\left[-\\frac{(m-b)^2}{2 a^2}\\right]$,\n",
    "\n",
    "and is charaterized by two parameters, the location $b$ of its peak, and its width $a$ (rigorously, this width is defined as the standard deviation of the curve). Because of the way Gaussians look, they're often also known as Bell curves:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gauss(t,a,b):\n",
    "    exponent = -(t-b)**2\n",
    "    exponent /= 2*a**2\n",
    "    return np.exp(exponent)/np.sqrt(2.*np.pi*a**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t_coords = np.linspace(-10.,10.,1025)[:-1]\n",
    "\n",
    "f, ax = plt.subplots(figsize=(9,6))\n",
    "ax.plot(t_coords,gauss(t_coords,0.5,0.),lw=2,c='red')\n",
    "ax.set_xlabel(r\"$t$\")\n",
    "ax.set_ylabel(r\"$g_t$\")\n",
    "ax.set_ylim(-0.2, 1.2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Q1. What is the Fourier transform of a Gaussian that is centered on the origin?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Q2. Suppose we want to make $g(t)$ more squished together.  How would we change $a$?  What happens to its Fourier transform $\\tilde{g}(f)$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Solution:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Q3. Suppose we change the value of $b$ away from zero (to say 2).  What happens to $g(t)$? What happens to its Fourier transform $\\tilde{g}(f)$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Solution:"
   ]
  },
  {
   "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}