{
 "metadata": {
  "name": "notes_fourier_kernel"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "The Discretization Kernel"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We've shown that the discrete Fourier transform\n",
      "$$\n",
      "  \\widehat{h}_k = \\sum\\_{i=0}^{N-1} h(t_0+j\\delta t)\\\\,e^{-i2\\pi jk/N}\n",
      "$$\n",
      "and the continuous Fourier transform\n",
      "$$\n",
      "  \\widetilde{h}(f) = \\int_{-\\infty}^{\\infty} h(t)\\\\,e^{i2\\pi f(t-t_0)}\\\\,df\n",
      "$$\n",
      "are related by\n",
      "$$\n",
      "  \\widehat{h}\\_k = \\int\\_{-\\infty}^{\\infty}\n",
      "  \\Delta\\_N([f_k-f]\\delta t)\n",
      "  \\\\,\\widetilde{h}(f)\\\\,df\n",
      "$$\n",
      "where\n",
      "$$\n",
      "  \\Delta_N(x) = \\sum_{j=0}^{N-1} e^{-i2\\pi jx}\n",
      "  =\\begin{cases}\n",
      "    N\\\\ , & x\\in\\mathbb{Z} \\\\\\\\\n",
      "    e^{-i\\pi (N-1)x} \\left(\\frac{\\sin \\pi Nx}{\\sin \\pi x}\\right)\n",
      "    \\\\ , & x\\notin\\mathbb{Z}\n",
      "  \\end{cases}\n",
      "$$\n",
      "and in particular that the periodogram for wide-sense stationary data is related to the PSD by\n",
      "$$\n",
      "\\langle|\\widehat{h}\\_k|^2\\rangle\n",
      "= \\int\\_{-\\infty}^{\\infty} |\\Delta_N([f\\_k-f]\\delta t)|^2\\\\,S\\_h(f)\\\\,df\n",
      "$$\n",
      "So let's get a handle on $|\\Delta_N(x)|^2=\\left(\\frac{\\sin \\pi Nx}{\\sin \\pi x}\\right)^2$ by plotting it in matplotlib.\n",
      "\n",
      "First, let's plot it for $N=16$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "from __future__ import division\n",
      "rcParams['figure.figsize'] = (10.0, 8.0)\n",
      "rcParams['font.size'] = 14\n",
      "x=linspace(-1.5,1.5,5e3)\n",
      "D16 = (sin(16*pi*x)/sin(pi*x))**2\n",
      "plot(x,D16,'k-',label=r'$N=16$')\n",
      "legend()\n",
      "xlabel(r'$x$')\n",
      "ylabel(r'$|\\Delta_N(x)|^2$')\n",
      "yticks(arange(5)*64)\n",
      "ylim([0,300])\n",
      "grid(True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We see that $|\\Delta\\_N(x)|^2$ is indeed periodic, with period 1, and has the value $N^2=16^2=256$ for integer $x$.\n",
      "It's also pretty sharply peaked at integer values, and if we zoom in and compare $N=16$ to $N=64$, we see that it becomes more sharply peaked as $N$ gets larger:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D64 = (sin(64*pi*x)/sin(pi*x))**2\n",
      "plot(x,D16,'k-',label=r'$N=16$')\n",
      "plot(x,D64,'b-',label=r'$N=64$')\n",
      "legend()\n",
      "xlabel(r'$x$')\n",
      "ylabel(r'$|\\Delta_N(x)|^2$')\n",
      "yticks(arange(5)*1024)\n",
      "ylim([0,4200])\n",
      "xlim([-1./8,1./8.])\n",
      "grid(True)\n",
      "tickvals=arange(-3,4)/32.\n",
      "ticklabs = [(r'$%d/32$' % x) for x in range(-3,4)]\n",
      "xticks(tickvals,ticklabs);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So in the limit of large $N$, $|\\Delta\\_N(x)|$ is, up to a constant, an approximation for an infinite sum of Dirac delta functions (what Gregory calls a \"bed of nails\")."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}