Approximate_Periodic_Function.ipynb

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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The claim: \n",
      "==========\n",
      "----\n",
      "\n",
      "## any function can be approximated as a series of sines and cosines, or \n",
      "\n",
      "\n",
      "## $f(x) = \\sum_{n=1}^{\\infty} a_n \\cos(\\frac{2\\pi n x}{P}) + b_n \\sin(\\frac{2\\pi n x}{P})$, where $P$ is the period of function $f(x)$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Initialization\n",
      "==============\n",
      "----"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# automagically display plots\n",
      "%pylab inline\n",
      "\n",
      "# load useful libraries\n",
      "import numpy as np\n",
      "from scipy import signal\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "WARNING: pylab import has clobbered these variables: ['f']\n",
        "`%matplotlib` prevents importing * from pylab and numpy\n"
       ]
      }
     ],
     "prompt_number": 195
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# function that takes coefficients (const, sin, cos)\n",
      "# and returns array of y-values\n",
      "def makeFunction(const_coef, sin_coef, cos_coef, xVal_arr):\n",
      "    # initialize yVal_arr\n",
      "    yVal_arr = np.zeros(len(xVal_arr));\n",
      "    \n",
      "    # first the sines\n",
      "    # loop over entries from the dictionary\n",
      "    for coeff, val in sin_coef.items():        \n",
      "        # determine the period of the sin/cos\n",
      "        # notice how we turn the variable 'n' into an integer!\n",
      "        n = int(coeff[1:len(coeff)]); \n",
      "        # print coeff, n, val;\n",
      "        \n",
      "        # add appropriate function to return value\n",
      "        yVal_arr += val*np.sin(xVal_arr * 2 * np.pi * n);\n",
      "        # print yVal_arr;\n",
      "        \n",
      "    # now the cosines\n",
      "    # loop over entries from the dictionary\n",
      "    for coeff, val in cos_coef.items():        \n",
      "        # determine the period of the sin/cos\n",
      "        # notice how we turn the variable 'n' into an integer!\n",
      "        n = int(coeff[1:len(coeff)]); \n",
      "        # print coeff, n, val;\n",
      "        \n",
      "        # add appropriate function to return value\n",
      "        yVal_arr += val*np.cos(xVal_arr * 2 * np.pi * n);\n",
      "        \n",
      "    # and finally, the constant offset\n",
      "    yVal_arr += const_coef;\n",
      "    \n",
      "    return yVal_arr;   \n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 221
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "# t is an array of the x values\n",
      "t = np.linspace(-1, 1, 500, endpoint=False)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 222
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Pick a function; any function.  Can we match it?\n",
      "================================================\n",
      "----"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# choose any function\n",
      "# a square pulse, for starters\n",
      "f = signal.square(t * 2 * np.pi );\n",
      "\n",
      "# and plot\n",
      "plt.plot(t, f);\n",
      "plt.ylim(-2, 2);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7f4883e5e4d0>"
       ]
      }
     ],
     "prompt_number": 219
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Building a function from sines and cosines: Guessing\n",
      "====================================================\n",
      "----"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# type 'Shift-Enter' to evaluate this cell\n",
      "\n",
      "# /////////////////////////\n",
      "# set the coefficients here\n",
      "# /////////////////////////\n",
      "const_coef = 0.5;\n",
      "cos_coef = {'a1':0.5, 'a2':1.0};\n",
      "sin_coef = {'b1':1, 'b2':0.2, 'b3':2, 'b7':2};\n",
      "\n",
      "\n",
      "\n",
      "# ///////////////////////////\n",
      "# this will plot the function\n",
      "# ///////////////////////////\n",
      "y_arr = makeFunction(const_coef, sin_coef, cos_coef, t);\n",
      "plt.plot(t,y_arr);\n",
      "plt.plot(t,f,'k',linewidth=3.0,alpha=0.65);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Pfga85z3m7+10mrIyoKKiP5L621Ggug4YsCuot7cPvOXUk5wkdQirT11tAsBt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\ncXrPMWMy9z3wwAN6jR+kpIK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       "text": [
        "<matplotlib.figure.Figure at 0x7f4883f7a850>"
       ]
      }
     ],
     "prompt_number": 230
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}