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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Übung 11 - Abtastung & Aliasing\n",
+    "\n",
+    "In dieser Übung betrachten wir ein Signal $x(t)=\\cos(2\\pi\\cdot f\\cdot t)$ mit der Frequenz $f=10\\,\\text{Hz}$ , welches mit einer Frequenz $f_T$ abgetastet werden soll.\n",
+    "\n",
+    "Um die bei diesem Unterfangen möglicherweise auftretenden Probleme zu visualisieren, setzen wir in diesem Notebook eine variierende Abtastfrequenz $f_T$ ein, die durch einen Slider variiert werden kann.\n",
+    "## Berechnung und Definition"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from numpy import cos, pi, abs\n",
+    "import numpy\n",
+    "from scipy.fft import fft, fftshift\n",
+    "\n",
+    "\n",
+    "f = 10 # Hz\n",
+    "ft = 40 # Hz\n",
+    "duration = 1 # s\n",
+    "\n",
+    "t = numpy.linspace(0, duration, 5000)\n",
+    "n = numpy.arange(0, duration*ft)\n",
+    "zeropadding = 2**10\n",
+    "Omega = numpy.linspace(-pi, pi, zeropadding)\n",
+    "\n",
+    "def x_continuous(t, f):\n",
+    "    return cos(2*pi*f*t)\n",
+    "\n",
+    "def x_discrete(n, ft):\n",
+    "    return cos(2*pi*f/ft*n)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Visualisierung"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "scrolled": false
+   },
+   "outputs": [],
+   "source": [
+    "from IPython.display import display\n",
+    "from ipywidgets import IntSlider, FloatSlider\n",
+    "from matplotlib import pyplot\n",
+    "%matplotlib notebook\n",
+    "\n",
+    "ft_slider = IntSlider(min=1, max=f*5, layout={'width':'100%'}, value=2)\n",
+    "\n",
+    "figure, (axis1, axis2) = pyplot.subplots(2, 1)\n",
+    "pyplot.tight_layout()\n",
+    "figure.set_figwidth(13)\n",
+    "figure.set_figheight(13)\n",
+    "x_result = x_continuous(t, f)\n",
+    "\n",
+    "def slider_update_plot(changes):\n",
+    "    ft = ft_slider.value # get the sampling frequency from the slider\n",
+    "    n = numpy.arange(0, duration*ft)\n",
+    "    dft = abs(fftshift(fft(x_discrete(n, ft), zeropadding)))/ft/duration\n",
+    "    max_index = numpy.argmax(dft[0:int(zeropadding/2)])\n",
+    "    omega_max = abs(Omega[max_index])\n",
+    "    \n",
+    "    # visualize\n",
+    "    axis1.cla()\n",
+    "    axis1.set_xlim((0, 1))\n",
+    "    axis1.plot(t, x_result)\n",
+    "    axis1.plot(t, x_continuous(t, omega_max*ft/(2*pi)))\n",
+    "    axis1.stem(n/ft, x_discrete(n, ft))\n",
+    "    axis1.set_xlabel('$t$ in s', fontsize=22)\n",
+    "    axis1.set_yticks([-1, 0, 1])\n",
+    "    axis2.cla()\n",
+    "    axis2.set_xlim((-12, 12))\n",
+    "    axis2.set_ylim((0, 1.1))\n",
+    "    axis2.plot(Omega*ft/(2*pi), dft)\n",
+    "    axis2.set_xticks(numpy.arange(-12, 13, 1))\n",
+    "    axis2.set_ylabel('DFT{x[n]}', fontsize=22)\n",
+    "    axis2.set_xlabel('$f$ in Hz', fontsize=22)\n",
+    "\n",
+    "ft_slider.observe(slider_update_plot)\n",
+    "slider_update_plot(None)\n",
+    "display(ft_slider)"
+   ]
+  }
+ ],
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