fourier transform.ipynb

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  {
   "cell_type": "markdown",
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   "source": [
    "# Going from Fourier Series to Fourier Transform"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Last time we defined the Fourier Series as\n",
    "\n",
    "$$\n",
    "f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} a_n \\cos(nx) + b_n \\sin(nx)\n",
    "$$\n",
    "\n",
    "### Let's look at the square wave centered at $x=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
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   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import numpy as np\n",
    "import scipy.signal as signal\n",
    "import matplotlib.pyplot as plt\n",
    "pi = np.pi\n",
    "x = np.linspace(-3*pi, 3*pi, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
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   "source": [
    "plt.axhline(0, color='gray', lw=1)\n",
    "plt.plot(x, 0.5 + 0.5 * signal.square(x + pi/2), lw=1.5, \n",
    "         label=r'$f(x)=\\frac{1}{2} + \\sum_{n=1,3,5\\ldots}^{\\infty}a_n \\cos(nx)$')\n",
    "plt.yticks([-1, 0, 1, 2], ['$-1$', '$0$', '$1$', '$2$'])\n",
    "plt.xticks([-3*pi, -2*pi, -1*pi, 0, pi, 2*pi, 3*pi], ['$-3\\pi$', '$-2\\pi$', '$-\\pi$', '$0$', '$\\pi$', '$2\\pi$', '$3\\pi$'])\n",
    "plt.xlim(-3*pi, 3*pi)\n",
    "plt.ylim(-0.5, 2)\n",
    "plt.legend(fontsize=16, fancybox=True, framealpha=0.3, loc='best')\n",
    "plt.rcParams['figure.figsize'] = (11, 4)\n",
    "plt.rcParams.update({'font.size': 16})\n",
    "plt.title('Fourier Series of a Square Wave', fontsize=14)\n",
    "plt.xlabel('$x$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### But we didn't show how to calculate $a_n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Let's change our Fourier Series Equation\n",
    "\n",
    "$$\n",
    "f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} a_n \\cos(n \\omega_0 x) + b_n \\sin(n \\omega_o x)\n",
    "$$\n",
    "\n",
    "### where\n",
    "\n",
    "$$\n",
    "\\omega_0 =\\frac{2\\pi}{T}\n",
    "$$\n",
    "\n",
    "### and $T$ is the fundamental period (for the square wave above, $T=2\\pi$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The equation for $a_n$ is\n",
    "$$\n",
    "a_n = \\frac{2}{T}\\int_{-T/2}^{T/2} f(x)\\cos(n \\omega_0 x)dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### For the square wave above, the limits are only $-T/4$ to $T/4$ because $f(x)=0$ for the rest of the wavelength."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "a_n = \\frac{2}{T} \\int_{-T/4}^{T/4}\\cos(n \\omega_0 x)dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Because $\\cos(nx)$ is even, we can reduce the limits to $0$ to $T/4$ and multiply by $2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "a_n = \\frac{4}{T} \\int_{0}^{T/4}\\cos(n \\omega_0 x)dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "a_n = \\frac{4}{n \\omega_0 T} \\sin(n \\omega_0 x)\\bigg|^{T/4}_0  =\\frac{4}{n \\omega_0 T}\\sin \\left( n\\omega_0 \\frac{T}{4} \\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "a_n =\\frac{2}{n\\pi}\\sin \\left(n \\omega_0 \\frac{\\pi}{2} \\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "plt.axhline(0, color='gray', lw=1)\n",
    "plt.plot(x, 2/(x*pi) * np.sin(x * pi/2), '--', label=r'$\\frac{2}{n\\pi}\\sin \\left( n\\omega_0 \\frac{\\pi}{2} \\right)$')\n",
    "an = [2/(n*pi) * np.sin(n * pi/2) for n in range(1, 10)]\n",
    "plt.plot(0, 1, 'bo')\n",
    "plt.plot(range(1, 10), an, 'bo')\n",
    "plt.yticks([-1, 0, 1, 2], ['$-1$', '$0$', '$1$', '$2$'])\n",
    "plt.xticks(list(range(9)),\n",
    "          ['$0$', '$\\omega_0$', '$2\\omega_0$', '$3\\omega_0$', '$4\\omega_0$', \n",
    "           '$5\\omega_0$', '$6\\omega_0$', '$7\\omega_0$', '$8\\omega_0$'])\n",
    "plt.xlim(0, 8)\n",
    "plt.ylim(-0.5, 1.5)\n",
    "plt.legend(fontsize=16, fancybox=True, framealpha=0.3, loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### What if we make the squares farther apart by making $T=4\\pi$ ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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    "plt.axhline(0, color='gray', lw=1)\n",
    "x = np.linspace(-4*pi, 4*pi, 1000)\n",
    "f = (0.5 + 0.5 * signal.square(x + pi/2)) * (0.5 + 0.5 * signal.square(x/2 + pi/2))\n",
    "plt.plot(x, f, lw=1.5)\n",
    "plt.yticks([-1, 0, 1], ['$-1$', '$0$', '$1$'])\n",
    "plt.xticks([-4*pi, -3*pi, -2*pi, -1*pi, 0, pi, 2*pi, 3*pi, 4*pi],\n",
    "          ['$-4\\pi$', '$-3\\pi$', '$-2\\pi$', '$-\\pi$', '$0$', '$\\pi$', '$2\\pi$', '$3\\pi$', '$4\\pi$'])\n",
    "plt.rcParams.update({'font.size': 16})\n",
    "plt.xlim(-4*pi, 4*pi)\n",
    "plt.ylim(-0.5, 1.5)\n",
    "plt.rcParams['figure.figsize'] = (11, 4)\n",
    "plt.title('Pulse wave', fontsize=14)\n",
    "plt.show()"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The new fundamental frequency becomes $\\omega'_0 =\\omega_0/2$ \n",
    "### The $a_n$ coefficients become\n",
    "\n",
    "$$\n",
    "a_n = \\frac{4}{n\\omega'_0 T} \\sin \\left(n\\omega'_0\\frac{T}{8}\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "a_n = \\frac{2}{n\\pi} \\sin \\left(n \\omega'_0 \\frac{\\pi}{2} \\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Since the fundametal frequency is cut in half, the spacing between coefficients are closer, but bound by the same envelope"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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       "<matplotlib.figure.Figure at 0x1b9924d1908>"
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.axhline(0, color='gray', lw=1)\n",
    "plt.plot(x, 2/(x*pi) * np.sin(x * pi/2), '--')\n",
    "nprime = np.arange(0.5, 10, 0.5)\n",
    "an2 = [2/(n*pi) * np.sin(n * pi/2) for n in nprime]\n",
    "plt.plot(0, 1, 'go')\n",
    "plt.plot(nprime, an2, 'go', label=r\"$\\frac{2}{n\\pi}\\sin \\left( n\\omega_0' \\frac{\\pi}{2} \\right)$\")\n",
    "plt.yticks([-1, 0, 1, 2], ['$-1$', '$0$', '$1$', '$2$'])\n",
    "plt.xticks(list(range(9)),\n",
    "          ['$0$', '$\\omega_0$', '$2\\omega_0$', '$3\\omega_0$', '$4\\omega_0$', \n",
    "           '$5\\omega_0$', '$6\\omega_0$', '$7\\omega_0$', '$8\\omega_0$'])\n",
    "plt.xlim(0, 8)\n",
    "plt.ylim(-0.5, 1.5)\n",
    "plt.legend(fontsize=16, fancybox=True, framealpha=0.3, loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### If $\\omega_0' \\rightarrow \\omega_0/10$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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     "metadata": {},
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   ],
   "source": [
    "plt.axhline(0, color='gray', lw=1)\n",
    "plt.plot(x, 2/(x*pi) * np.sin(x * pi/2), '--')\n",
    "n = 0.1\n",
    "nprime = np.arange(n, 10, n)\n",
    "an2 = [2/(n*pi) * np.sin(n * pi/2) for n in nprime]\n",
    "plt.plot(0, 1, 'go')\n",
    "plt.plot(nprime, an2, 'go', label=r\"$\\frac{2}{n\\pi}\\sin \\left( n\\omega_0' \\frac{\\pi}{2} \\right)$\")\n",
    "plt.yticks([-1, 0, 1, 2], ['$-1$', '$0$', '$1$', '$2$'])\n",
    "plt.xticks(list(range(9)),\n",
    "          ['$0$', '$\\omega_0$', '$2\\omega_0$', '$3\\omega_0$', '$4\\omega_0$', \n",
    "           '$5\\omega_0$', '$6\\omega_0$', '$7\\omega_0$', '$8\\omega_0$'])\n",
    "plt.xlim(0, 8)\n",
    "plt.ylim(-0.5, 1.5)\n",
    "plt.legend(fontsize=16, fancybox=True, framealpha=0.3, loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Fourier Series becomes the Fourier Transform when\n",
    "$$\n",
    "T \\rightarrow \\infty, \\qquad \\omega_0 \\rightarrow 0\n",
    "$$"
   ]
  }
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