Lessons/FourierAnalysis/1_DiscreteSampling.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# An Interesting Observation Regarding Discrete Data"
   ]
  },
  {
   "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 notebook, we make an interesting observation about dealing with sines and cosines of discrete data. Let's begin by plotting\n",
    "\n",
    "$g(t) = 5 \\sin \\left(\\frac{2 \\pi t }{ T } \\right)$, with $T=0.5$\n",
    "\n",
    "To make this look like a nice continuous function, we typically pick a pretty fine grid of $t$ values to plot. Let's go from $t=0$ to $t=1$, spread over $129$ points.\n",
    "\n",
    "Notice that $T=0.5$ corresponds to a frequency of $f=1/0.5 =2$ Hz."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.linspace(0., 1.0, 129)\n",
    "g_fct = 5. * np.sin(2. * np.pi * t / 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, '$g(t)$')"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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,g_fct,lw=2,c='red',marker='o')\n",
    "ax.set_xlabel(r\"$t$\")\n",
    "ax.set_ylabel(r\"$g(t)$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Q1. Warm-up exercise: Make a similar plot but with just 33 points along the $t$ axis."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Q2. Now do the same (with 33 points along the $t$ axis) but for $T=1/66\\approx 0.015$.  Notice that since this period is much *shorter* than $T=0.5$, the frequency of the oscillation should be *faster*.  Plot the new set of points on top of the old ones, but in black and using \"x\" markers instead of circles. Think about what you expect this to look like (how much faster should the oscillation be?), and then make the plots. Warning: counterintuitive result coming up!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Q3. Make sense of what's going on by repeating the last problem but with $129$ points along the $t$ axis. Can you explain what happened?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b> Solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "hera",
   "language": "python",
   "name": "hera"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
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