{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Continuous Signals\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Standard Signals\n",
    "\n",
    "Certain [signals](https://en.wikipedia.org/wiki/Signal_%28electrical_engineering%29) play an important role in the theory and practical application of [signal processing](https://en.wikipedia.org/wiki/Signal_processing). They emerge from the theory of signals and systems, are used to characterize the properties of linear time-invariant (LTI) systems or frequently occur in practical applications. These standard signals are introduced and illustrated in the following. The treatise is limited to one-dimensional deterministic time- and amplitude-continuous signals."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Complex Exponential Signal\n",
    "\n",
    "The complex exponential signal over time $t$ is defined by the [complex exponential function](https://en.wikipedia.org/wiki/Exponential_function#Complex_plane)\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = e^{s t} \n",
    "\\end{equation}\n",
    "\n",
    "where $s = \\sigma + j \\omega$ denotes the complex frequency with $\\sigma, \\omega \\in \\mathbb{R}$ and $j$ the imaginary unit $(j^2=-1)$. The signal is often used as a generalized representation of harmonic signals. Using [Euler's formula](https://en.wikipedia.org/wiki/Euler's_formula) above definition can be reformulated as\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = e^{(\\sigma + j \\omega) t} = e^{\\sigma t} \\cos(\\omega t) + j e^{\\sigma t} \\sin(\\omega t)\n",
    "\\end{equation}\n",
    "\n",
    "The real/imaginary part of the exponential signal is given by a weighted cosine/sine with angular frequency $\\omega = 2 \\pi f$. For $t>0$, the time-dependent weight $e^{\\sigma t}$ is\n",
    "\n",
    "* exponentially decaying over time for $\\sigma < 0$,\n",
    "* constantly one for $\\sigma = 0$,\n",
    "* exponentially growing over time for $\\sigma > 0$,\n",
    "\n",
    "and vice-versa for $t<0$. The complex exponential signal is used to model harmonic signals with constant or exponentially decreasing/increasing amplitude."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following example illustrates the complex exponential signal and its parameters. The Python module [SymPy](http://docs.sympy.org/latest/index.html) is used for this purpose. It provides functionality for symbolic variables and functions, as well as their calculus. The required symbolic variables need to be defined explicitly before usage. In the example $t$, $\\omega$ and $\\sigma$ are defined as real-valued symbolic variables, followed by the definition of the exponential signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{t \\left(1.0 i \\omega + \\sigma\\right)}$"
      ],
      "text/plain": [
       " t⋅(1.0⋅ⅈ⋅ω + σ)\n",
       "ℯ               "
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "sym.init_printing()\n",
    "\n",
    "t, sigma, omega = sym.symbols('t sigma omega', real=True)\n",
    "s = sigma + 1j*omega\n",
    "x = sym.exp(s*t)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now specific values for the complex frequency $s = \\sigma + j \\omega$ are considered for illustration. For this purpose a new signal is defined by substituting both $\\sigma$ and $\\omega$ with specific values. The real and imaginary part of the signal is plotted for illustration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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XEK4XNogSSmg0RyhOrfeLYJBFyFwoVnHfvqADrOI9rGaDrRzv9jGIEmCLcjlU3iu0uxcZZBHgrXD0sd3y3kZ3kUWAvmny6WnIxZvwjiwCHLCuvwMnfM9VHqIEaNUxRQwXIYaLYJBFaJU7WhRzqF8d2yBKgOckY5baWezqTTDIImDOamtAzXS16juwijNSQTQINzKuJnoHtPio1GmiEWLv90g2iBJmqkH2qWgU1ZwOwSCLMDK3XyCmfROvFzAICZxJQ1MlVfPpRUauhWMc4gVE9F2uNpziBlkEhn6KRuZuVY0X4R1RQsK8g6kNc0RkXO1VIYMoodClphSjY9R+yr8DWhxmD1coM3t966Ge8gZRQp+RodOZ0dtnz9FFlABjjHssmHXRAfL5ZA5CAifnKHuRHZ0gj2MgOIgS0F9pNxQqq8hlw00wCAkwFyoc3Towq7S9Wu7IWZgxKXWiln1iiISzJuMhIODtMyzrmV6JZiy8vN2mLkQKnQiu5ScJXYbV/7yBh5DQYReirvmFGQg/lOp4WtWFSOFGLZxBPFuXkMOe911IKW1ECVjkwmPPMV0UC4FSYOOiT0ZayIXdsj6hQD6kFBiuNMa4ssfokHgYBlFCaQWKWcTQnmE7NC6kFHq1bEca7vjv/RVdSCmoYZWf6hj77VmxdRElYAAXlaJv1n69h0GUwCjZUUWOJ49wvYeFSKGbMmrSnxoj7PAyH1IK7Faul1E+ufU3DsVCiwLVxV2mKVFrPV1PsZBSuCXdVB4bXKl8KBZalI5JrYo8leObeIgSsnjalMIBmXsVzkMWAe5p0wVgTtDbd3AhpRS4z2391IQ5dbWWhRZlpCD9mgcI0IhXvSyklArfbi55qWlPSy60KAMW8FR5Le1qLYMooUHNz0ecRrkYFloUKMv+PHsbjQ6gxXsOc32mksIo11tYaFFgB8y62jCMdD3DQkqhr1O0spjA4t1PLLQojaGBqzukfjPeESVAL3XtbnXGo08ssIrXLhvClGOOvFvKQqRAe1Pd65PhrMa2tYMHLQpXsvSnAg/yzItiIKXAW2pLbUx0hdN7PWhRapUQCKoNiO+nWEgpqczAvXYqpwTHvhyKhZSSubGimibDhdq7AS6klMJ48Cl6GTMffPdDsZBS6gg9NJFXxrL3Q7GQUtqUACzKMcGe6cpBlDBDZrgDpyQMonQRDEICA1GGzMcTJidbcY8SD1IKd25EDh87naVaD1FCD13acExuKB272IVIgfkYMGB4GAFmVaa+fSgeRApnVDQ8fqrBCZtz7N05FwKlcgsO031+wfZCEZjQT1dxIVIa7Vq4d43xJC3ud3cALc5NAYwy/GC/Nxk9hAT0y84u8yoyGcfdVB5CAlfwysS7oWEy7Le9FuohSsDXwVvR6uSv7OHhIUqATkKnfHH84OvO6wkGUUIbnV1Ypsqxg7FcRAlDTBtuM8OsL/UiGEQJE8Z5l33pOMKZzj2EBIatDk5WmScU8Gmfoechi0B2pbjBcdhLJh6iBIk0FHHFRLEVlYcswhhRHwzztvV0Ed4RJeRc6FtDDPOgbMPdQxZh9MG1Dka8wn8uF+EdUQJnzyjixA97tZJBFmFw61rXRCV+4hDeESXAp8pci4A4wfy/mtUgizBmCfLg0KGzr2Y1iBLQX2Bjibjg017NapBFmIVrcRTz3NT10gZRQof/WOXzBExbe+HEQxaBMTEiDW3H/HuAFh90yZ9fCe16ZYMsQuMRWBXPHcPvIosA53lIfwk5tX59NoMoYXK7R9a+Q+VQPwSDLAKcgil9Pgz6eRfhHQGhBS6mSoypDMi8zUEPWYQ2RpPfiY1HFi/CO6IEdCtG9UkXbjti20UWoZWqNYXrebaMPEQJsEJlc1ODbPaI9pBFaF1OKGC4oDdve8hDlJATXDXpwjSr6in/DqziPIMlvwJ3Yp/VcBEllJZ4GozKcOa9QOYAWhyaIHCGCYzFrfc3MIgSGly/ruocrsC8voFBlDBCKhgi9DlnKHv3wEOUINuM8SUOEfzz6xsYhASeeIblFTmHlQnHabepgygBhkHPunFEb74cgkGU0AdPMjHamgbrHv8eQgJsu5gj5kaYBtwq3HrbQ0go6MCdhwI6z6K3tH1oDyGh4ncqzW26AXul2IhRlLH0tMa5KJQ/M33Mp0VdiBQ8qqMPQl7lqGXeH8GFSCl48pQYNslF0PYqi4d8k5wKpUyGmsGMgH0GZfioJBdSyohZqjskGDjv5WUXIgVGHfQtf2oUOYuyt1pcSCmFVhSDxxoMMKnIplhIKS3g31kWuAJPhOdDsZBSGBhIc14CHUfdn9yFlDJrUDmj2jAT1kOxECk8YAkvQ+WTsbq7xTxIKZjtufgEe54WXm2HYRAlQFCk6aG6ZuP5vc2wkFK45x7Tchq4Z38oFlIKhnzuXeQSc3O9iYWUUkPo4jEFHoBL2+92oUWpsvUsu+kwYsJVMQsppWEW5NMhxyx0LCIXWpQOzddUDstnXm1sIaVwA5fDDvIMiy9e72KhRRmZO44i7zzhfFEMpBTMzilpsxQmH7ka2UKLAmeMhz8ox0v26/UtpBRYHBJ8K/J59tJcaFGGbBYs+bi6mEFIkCX2HLRRCv7e+wgutCiYRZv+EiaLsn11F1JKLCG1pxljuxgGUQKagYtVIk+1nNHlQYtSU0559Ygar2cYZBG4/rK6EH7oaEkPUkoulQGj0rfrzEcZedCiTFmwkhHU+1mocCGlwPuTzQaO0zHq9n9cSClQa7KhQW0w8gmldaFFgcUYVU31IYf6D8VASmklFRUz9vFmGEQJ3EluWt1cQ7kIBlHCwBhdP0RT5+4pFlIKT4yzP0iA+DyxWy5ECp0c2PJF5Lm21na9PEgp3OFOOj3VisqPQ7GQUvI668NTP7XH6yEGUQLMls5ex/WeUXYouw8pBW4PhhnkWdrnfhMLgTIC81FEifkOurD4NLELkUJzL0aG9dM0yTzb+lA8iBQxwrgJiV7BaWfuweVCpNAbZr6qVOmbnPHrACg+GbvOvTKx93p+zmV4AItnHvqt8CNhKQaYDXupxkOUMNAicJ2ZwaPC2O2HYBASuFxXoZlosvLM4Q7Z8hAlVMY5v2oUJbitUwfQ4r0Onv6lQcnkTu2UN4gSoFFALrKmUfZ3dgAWr588l495tGCohRm3eeIhSkhMaZUpTnGfGfUALZ554jRzWQ+dcbTr9w2iBHg9jAEuTGY24/Y/PEQJsFYSD3HwLE8q86qRQRaBAVO60lgY6HAR3hElNI69SjEjv843cxAlYFLrmKIKD5gxtvEQDKIEzLMVLnphcHK8G+kdWMUbWqCLNM6wF+M8RAmTU438DPrMWTv2kEUokYehKL7VhYeQwOP1BaaGiGObZ2Q6yCIw9Eh/B59pR3d7yCLAXGziTTKart+Ed0QJ4hBHFee441s8ZBGgPHNa4rP76CFKSCHSm9AH96NeHGQReORAHzyvzUoPWQR89ipi2OHH4vYQJWSG3jURl1m3I+Ahi9AjIxso5hG0q/wboMW5XR/ksXArUrl+3yCLgPep0hJzXFunHqKEygwMMggDzN37sxlECTw7PHRMMf/h1fUMsgjo56pIcuhfnmAQJfRO/5/igtn1fgeDKIGJMNBLqKpC2GdwXUQJmCCCvhoDkI6adxASOvs8TbTCHfTY90zrIUrgXn6kdJY5z3e2gBYvgZtinC5a7v0ubxAlMKte5WwEa+gabBbQ4iBmTBKVe+457EUgDyFh8Eh5gb/EDAM01HeTOogSSkQn769aeaImnuJvchamcUnXmvtb6ABlh9p5yLcP5j3jiiTUa29c0GvPZqeLkMDTN5hakmRo4/LqM5R9CJQIBwyWGVORMd+pHJFfFBciheF3keZrajzeVOezX+ZDSumxNDlByCIpxotiIVIS13unLEFh1m+pPLrYh5TScmEumUSntJX5GBo+pBRqdRqZkgExP8dsXISETN+1Dw3dD5hyym4vD1JK4pFLzQo3GVxcDsVCSmECwBllMQ39NNbrKRZSCn0dnu0dDEicO6THh5QClV7lp3gGu52P4iCLMGhiixifII/rGRZSCncccxN55srrOBQLKQUzMoMLKUcnj+miWGhRWkqyAJUwoMtNeAO0+KQvNkSKl6uPM+hDi8IDc1nl8Hrq9eoWIqUwMQv31Cjn/vSheNCiMENTVTnPUJaLYiClRHj5KYl8pLmnbx9alA6lra04mFnieoqFlJLieDrE6MzGdSgWWhTNXkExN5PmxXhHlJADs0eofNaYrvay0KI0Jl1bcpiM7aIYSCklSAYifbosLmyKhRaFq87rBeNXxjuyCAMdIq9WjP3+JhZSSuVJ8iXnsYOLYqFFaZF5E7RH9EuveNCiaOIKilvZhxJcRAmYK/Lqp6js3R8NsghoiKi9ocFhbNdHtJBSoAMZlitjtGN4X69uIaWMVGUtlHqgMX78UCykFLiXvejTeZ4xHYZBFmFC9S2NxsM1V5e3ECl0Z2uWns24Mgl2WxQPUgrThBaVd3Ty1A7FQkpJsjggch65jOVQLKSUnGpcMwCUWsj9UCykFB7NXKlMUz0q1QJavHPzNMp82WLecVY+pJTJHf4im0yNsanXq1uIlMZ4VtnJooUY9ia6iyiB2ZO4INbl6PssF8NCSmGi2aAGCYyZvfXmQ6TA/OTaAvMvcUVmlrgpHkTKYMJLftvKpEq1HrPLQUiYGJaaGCbz5P49sDwIFDksFbj+xC1AJnh7li18iJQmmRdmeUXm4ayh7xnehUihacwI6YH+GlrYI9EBWJyxP42+QKedAN/+6bwe8k0yq+MPbl8ySzhXrZ6W8hAlVOY1gFg2Pcqz6eQiSuCx5VrkXADMmHARDEICT7dFfEwm6Yjo0ecJDqIE2DfcFqp0SFPZlryHKKEwB7H4D435Q075d0CL18FkQRRjTpnPIoGLKAFfhv2kckktPyvIHqDF4S5VaDn0yTnQPa/fN4gSmMCCDcdDI0xSfwgGUQJP6oZMcSt956xxERIYVAtDVsSVYbi7iRxkEars5kGMqX32dhHeESVgrIco4lz2qqIHrOIYUPz0hZuuY97l3xEloAuGrOI88v0GBlmEgiZLFNPW7vkivCOLMMsIUtHYUOWb8I4oIcNqavo7Oe9DJy6yCD1zfYXiWPL90gZRAlNAJ/mdMOeo11cwyCIw/538TmDW8XkR3pFFmJgKuoqZ6+YivCNKkADjuMTzKv8OrOLMHpxXPeO2ujxECdzsCdoSae8ieYAW5zUHQWoZO+yRccobRAn4kEMHSMo91usbGGQRoAiyDMHMXfGrRgZRAlP1VumPpZRj1niI3pzBULkmbwaLeh9Sc5FFgFaT7thyyLld5d8ALR4HXI1B8Wi9HP3lIErI8Geb6E1m8InzEAyy7v4IshkGMYMbjgZ2ECUwmwy8f2pmVPR643dAi7fKpK7U+xiKs12/bxAlDAaBycySkgQ1bIJBSEAfxJTITFLMZC15IxbBQZQAXUNXvzKssOwFfBdRAjN1V65a8Zz4eQULsDiDimA0Yfbl5g304X4DByGB8UGV0Y0M+6mygLgIDgICT02gubij16tu/z+PcCFSmKsHdWT+NMrp/T4UDyKFi9bMLvDif94D1ENIgClQqhwPq5J6fe+h+xApg4dRuYBEsxE6om3Hy4WUQtuUP8XE4p0BOYdiIaXAqWC+He6xwvQa2zp2IVKYKRimU5NcYFzGrrtiHqSUwuzxQ88rcjCkQ7GQUhqt6yq2eYQNcL6KByllpMDNDcrrdcTSh5QCL6OI01AYvZa3VnChReky0VI+J+bQ6ykWAqUGBvUGLg+x57UTauVDSuHeKY/tdZ7POBrdQ5QAv0HS4zF0D97Z/QwLLQrMeV1DrRiuex3TQ5TA8CMVN7zu9tY8ZBEwgOjyUg5jIF+Ed0QJJQ3JlNmrpE5/whR8aFF64dqDyEsZ43pxCymFi45cdqS8zZGvelloUTAC6CNTPlPoV2NZSCmNqlXEmIT3UUYXWQR0zLnkse/9HB9aFKiMOJccuuqqloWU0rOk1lY582AfioUWBbaYLNDLBJbuvmUhpQyGT62vxZ3JcigWWpSuK3xdMintGd1DlMBrKNanSvlq3jf5Ksy4irk6HBMyXOUNREr8ZDBxXr0697Mw40KL0occmZaRM/teV3UhpUQaTVrd0vM+wuJDSoHKYDSMqICRdkC3Dy1KgwOvP8VrT8r1FAspBZ2hrQ4UuFISD8VCSoEt2+SnCme/fr+LhRaFxqCq2cJlnn5RDKQUbu+LziwMh9uZ0XxIKcykyZBOzDIFvki4WsxCpMhhyhmyzGX1OpbjQ0pJE95dl8UhboKfIe9BSmlt0FxgDntUfc/YHkICJ8nBDISc+znB1v3yHkQK3bVEPnfxYgox7g/pQaA0XgpUGQTDY39c2n1exENIKJIsG3qAgYA97AP+LkJClesD8DuDij2nrYU8hIRGu3CW/mLe+BZ3CJGLKAF9p+CFeIuI7rFtgkFIYFJn9Lf8agzEkLt5FsFBlIBxQxOv8S6W1Lcn6yFKKMzB3xgOFju14CEYRAkMuUD3avi4s5Z6EQyihD5CQBvzgq54ki27iBJ4/gvGlCS9Pzl1XISEwaPMXDRtScLv9yKKhyghRu470RXgBXPllH8HtLhosEYxcwyNU/wd0OKZufezbJyjDU8/cpBFqEyCwIMm1xUPHqDFJaWKeDIYiuMq/g6s4nPy+iUefIEvOa/qG0QJteh3ZKaGdHU6B1kENHIWMT5KvdvfIEpoGIWwfSnOdSdIcJFFaJ2LzjLYW+vXOxhECZ3rO42Lqgwq2vrFQxaBu8tdxMzMmC/CO6KEEQrzZ1OMbxquz2CQRaiZB4kpzTFene4dWMVn5XVhKh7j6kUGUQIMaub01cfWvRnqIYuALzKLvljJ9xMMQgIjQmgjUDpz3BEWDqDFI6YxXl7Bbxn3IRMXWYQWeWyK4t6PpeIhSmCkqZzNZgjitp8cYBWvktOEYrhF4XoBgyghM+eXDiiezb7ewCCLgPfSF8uRu+kX4R1RQqmSCL7JGvo+j+AiSuCpzCJiJie+XuEd0OJMPNGSrnr0Uq4KGUQJMLihKEVj5pSuN34HtDjPVpcmGlkdrl3eIEqgBg+ikXmv1Li+sUFA6HQueSMZZ5UyeIneIniIEpirHG4+53loqe1/eIgSuPQ/ImdGJsDc63weQoJk7YTt9BKNu5vUyrUwFCYzJnBPC2Nvx414CAm03zKU5YsbrIOnSh6Cg5DAFYcWmIWT2cNT2NdT+BApkyukmSmOmII0cmp8KB4EysCcnuUABpdhJJb3eYoLkVLkPiDuSNdK+yY/N6D4kFJa4yFxGIxBLlrbk60LkQJzPzJxC+QMkDiR5T6kFLQ7Y7poC/aS92EQH1IKpj7eT0RrsMlRwkOxkFJ6rFyyaFwPbPvqBRdRAiYLHr4Ra62Onbzch0jhFW5MGC/yyoOg+yEepBTM3FnasTMLQzov70FK4QU7echSXZj4wvlQLKQUBjAGXd3LKYd8VcxCi8JaRpHXOPOeMFxIKfBB18NbK7VdDIMoAXapLG7JPXU57sHuQkqBZxS4mCIzL1zx6yEWWpSp35czadyHJV1ECXA+JT6Ec2mtV+/yIKWMGnmQSGbHeJn6LrQo0E7rpyTk4XoTCymFd30+8npyS/vQovQusRtNz+nvNSMXIoVHYLMkYaK8p7xXv1xoUXqS+0VVvi8f8SGlxBVFp0+Pp2IetCiYep+WLOcaRR9SSkq63E45z2qmQ7HQojAJ7Hp6iCdcwYWUwttpuIDKXsHMA9e7WGhRKkP4VZ7G3rP0ECVArQVRhUNy38zrTSy0KENX2zkgYNaW6yEWUkplKLUObebqzBfFQkpptHV0pMJLC6dTepBSYL20riOVhwp33JQLKWVIki96eSOc0CEH0OIzBI2pGXKGul9PsBApQy7kSC/1hHvdG5IeogRmuuSaKy+z5U1F8TAspBQm0cw619DNOB/Rg5TCM0OibKvOmle9LKQUJr2PGlDEFe69BORCpEj2x8CwJcmsFPJ2e11IKVzA4IId09DwlEI8FAuBwqO8tWteAsnftDMK+RApzKUWC02TLJF+x4h3IVIwgHKCB/2Kerop7T7sQt8+MoPlMVXGKZZZ6n2fevUhUmSVDgOIV+Vx6TU/BAuwuNyKA9uCeUKbpE5/yjuIEmaLTCrDrSKeaOuHYBASGPWdK+x0bq1hBD0rmC6iBOahgB3CqHrU87EJXEQJgyffXjTdMdbGKf4mZ2GNX6bXOuXqwbZ/3UGUQOsQ41I0+AkcdxElYATzcdRHdS+MeIAWL1y+TOIPwOSJ1+8bRAlMefsSS2qkJ17KkWthXgHR4Jzw8EBv7aqNQRaBy88iRnefTwyKiyihZzFvqIV4RPpqUIMoAT407SauBXJF+XoDgywCHyxi7liV6x0MooSZAr2TJveX5XrKvwOrOK/BTZSW3vZatouQAI8h83AzpYXOz1PeAqs4ryEqIo68L+Mq/44oITJ0Slo6w+eb5RAMsgi8gUrFXEC7Ce+IEhhukaaK095IdJFFQEOXpuLRc7wI74gSuC5c9NWgasNFMIgS4IdEeW7hsskp/iZfhRvzpYuUeu2qv0GUUKOEKPDLh327qQes4jzeJs1Wa5ln0DuIEhii0aSn8GBLvAgGWQS21RAxA1pvwjuiBBg3ssLII2Doi9c3NsgiTIne4Xit8DuuNjKIEgacev0wk6F7F8EgSpgJwyiqzql739dFSOA91ngziZbJmCHGJjiIEhirPOV38FlryYdgECVgbs+iBXn/Q2+n/DugxXkGEJ+Ei3t4xXb9vkGUUIfcltol0Vjp5RAMooQOZVxkFpqp7OB9FyGhc6OKs3lnNOgpbuRauITBqEzeMRzTzjvoIiSIBchTMkNy14yjfB2EBOauSDwPyDSgYe4tWxcBIcKnltuauTLEDdbdQh5CwuRae+P6S2FSBt6V+jA8CJQUuOuUmBhJwivkpJBSXIgUzecFL4drSbW1nd3Xh5RSYS3LKUJpnH17iQ8pRW6rymJhxpOAzEVISJoqUJarYqxljzkPUQKzGTCcpMktH/t+eR9SSm48aiiWMo8WhXQoFlJKKV0cStnMnv1iGEQJTPRS1BqHmxvqPAwLKYXZAXhjcWOIW9qHUH1IKZ26pIk8MyvjRbGQUkal/hU5N+Pr1V4WUgq6M68hprw2STuxKRZaFPQbBjfJGdW2w499iJTMa5EGO2qTu3B2kJEPKYXpzJj2l/J5bh7xoUUpM6wKo0+UbUu40KIwHEyfDm9p35LjQ0rBaAvrezFu8owUD1oU7pNp4/M05LzexUJKySXSKBQ55pfTjz1oUUatq/F5ZXO6KQZSSkmFboDK0w5u96FF6WnKna18OqNELoqBlII6xrFesc56NZhBlADfVY5UySr13Y89aFH0WIZ+4NrbTTGQUjr80LB63qz9aEkPWpTWRy5PZ20X4Q3Q4oPxWVpZOBQ7g4kPKQXtkKb2h9rDXu73oUWZRa7cliCA1r9QDEQKA+Xh3ag88QqPh+EgSoAF8vTsGGc+beVBSknMpJl0zSJLIutNsZBSmN23qcLt+I/tX7mQUtDYWfpc5OR0zCwXUkqPkoyWUwfDIEs6FAsphaEo/LxMlRviHBfFQqRUnhhiFByjdjAd1rkr5kFKgQ/JCBbOzfh4bTsGLkQKs2lyT+bFwFaGG7fdJT2IFN7oPSfXEIt8r1h3I3sQKdzDGnmFWMOevIwDDwIl85xB4HzDq1Nqq+kJNvIhUrrcyxmZkpNp3Z71Ww9g8SFZkdHw3Dxigvj9AAchgVvag7u6PMTPMymPEvYQJWDSo23ItGtTEixtgkGUMJj9K7+YBJF3GqVDMAgIhdZzYg5bpr+Ex/psiLmIEhLjx/qrV96oug/yuIgSqPnRFhJWfPK+u4gSapc0OIyoHvvybQ/Q4q2xz7wkfenYBzddRAm9dh7SkeOcZWfedhElQMeHIEttvCNzu7MeooQpeT0h5Wbns+rsASzOK+0iB0fnXVdQeLu8gyghZl7xQjFPD+xlJg9RQooSGAkx81ucN3AQJWReJjEprmXsvNsusghD7sNkeOQM+zCViygBXnSDcqC47mwPHqDFa8THySJlUHg/5Q2yCMyZNVWcWrkJ74gSoKp5kEHEeecadhEl9JS4LicV5eLqIRhkEWarUX9n9D36HUCLDznS3uVg0AhXi74Dq/iU9KAUVxr+V/l3RAnsjEl+h9mBt370kEWYmZqW4prS/QIGISFJKoopGZE7HZ/9DRxECbE2HnJlf2/w1uchGEQJiUHJ0n15VH62QzDIIozGLF4cgvPylz1ECQVdJIo4sxmvdzCIEuoQn63LNWgjjUMwiBI6owcG1SCD2vL1BIMoYTA0JVPce907sy5CAjycEbnYxlNQo6dDcBAlJP1DkvzWcvSFgyihyLQl0xFDWsohGEQJsD77Ct+Fbz/7IRiEBCrYzOPuTPAw0lnB8xAl4KMzupGZN+E6bbPMQ0hgRm6JsmF2wiF3Di+Cg4BQ5fq/xJt2JVat7id4CAmy2iu3CGUunEj+lMXwIFIYyyGGQSqSR35uPeZCpHTN5PriobHMrbrnW3iIEniPpqTD5Nkkqt/DsJBSBs9OT9moQ6fet376ECncP21FlmR4ejSlvZDvQkpJWe4VY9Q67LpY26FYSCkFbiZP5vDYOM9oj0OxkFIaT0XqglSBYxovioWUgtmYh1PlNF4OaV4UC5HCaydkUTfxw6FRnl1PH1oUXroxRY4PHc538SClwM5hHn3K4dWX0108SCncz6RTV3kWse1bgH1IKXhPrndRPhhwclXMQkqBM89ZmnIepbje3iBKwLeaDGWR63vO6oOHLEKXmwTp08Cl3xdY+pBSMPlPrjBQDv9kzwYupBRm++xL3q4tXxdSygiFR59F3nlzx6FYaFHGbLJGyuyCZV/97ENKmTBqwpInJqQ6FAuBwss7ot63VSWh516t8ZBFaLxYWMVQtPlmGEgpIEuqXfm6c8fR+NCijMKdbelCRc6lH4qBlJK4+pFWR83pfhULKQUKXc4AVe4lhH0kwIeUUlpjFIMMOp59vJ5iIaVU2I5c+eHQ5pVb7VAspBRemFq0wqGkPq/Xt5BSeBakJdE5g47k9RQLKYV30sqiQWW8Y7qfYiFSGBeOXyqiPyevK3+UngspRVLjZpHDXIjPqQIfUkrGWBU5PZJen2hkH1JKi0xrIdMHc5I80aY+pBQeemOS7Sqh3GfX2oVISUyYwWhxToU5nBRiPqSUmXlvqWyozHmsIg8hgdE4vAPvxSQ3ofB0zMPwIFKgD+BIwpOMPAuGnzof0oNAYfwHV6yZoIexrinvqAYXIgWOA5Q5el4MTLQQd5ZfHyJF82uhvSeD38M8DAchgbtBhcdiGfZJ1fCMLg9RwqTnBzuO+Vt2dlkPYHHmAgi1xtfIPOFwfAUPUQLcSTrTmnYw7bURD1EChnTFFDvwYkUOR2yCQZTA5KJTxIEHzuchGEQJmPgqw825Okgv7RAMQgIVPr8IpFAap7iRa+EYJdG0bErWs17qIYvAdCrxJbneWzgt5CBKgEXBwyi8KDmWsne6PEQJuXPHj+Ja+zgfzUGUUECPIsb/a3th2UOUUCsPTlKc47WX5CFK4CmNTimTqF0VepOvwlOOXFAK02P7OR6iBFpmQ2oZ0QHG9fsGWQReQCu15I54uL6xQZTAhKVZPg23e+L10QyyCDxKUEQcmaPtIrwjSpjMPaoPZsq8qx8ZZBGYP1ql58ozD2BxZrbIPLst9Ux1L7x6iBJiDphttSXmvgzVRZQAW5+rZ/px8umnDqIEniDK8mJM1TougkEWgbffsGtV9vyr+Fe5Fi68aEjaubRzEbqLKKF25jyTcMKws2p5gBZv+OxSxd7COB/YAlq8dzYCxZOxbNfLGkQJk/c6dKoPniAc1wMMQsKUVFBNwiMkLnl3agdRQo6S8mjIEfK+/QsPUULRvL3MzEiNdhEMooSGvp5EDNcxtHgIBlHC1DBCnqxBpffGiIeAIKqY+UckaHTMfV2HiygBs0obEDP+eR6X0kNIYKqAwmloMt6h7QSlLqKEkTprynt/57Ul4iEkJLm9NvN002TG7LRHjguRIiFntJ3l5ud0sg/6ECiTK/uSzVnyOcKl2Wa1C5ESmaRH1hgSb7PM20PwEBLEmJH8itw7wcvuLScXUgrzKzGKpBQJ19kOuwuRopfG0aYtTaIits3gQkrJaP6iyziFY6AfioWUAre6y09RHe0GdgAtTue16r4d0zjNU94gSoDxUXi8A3KMtZ2l0odIYTBmZCJ/yVUzTtJjH1IK9Ghn0AmzqUwmoD8UCymFEaJi/wdmkQ6neT1IKQUWoIrhne5rWlxkEXh12hCHgUcE2vUmFlJKldB+Sd6Rd5ZED9DiPI/GtBBMw1GgMOchWEgpPU05/8c0HPV4UR6iBHScJS070bMjX4V5LDqrdKYT4elCSpmMjFW3itcptuvjWYgUnpbLz0/Vin66X9uDFmVOSZ9COWbzbXq4kFJik+MMKu/xplhIKRiSktpWno4P1g7FQkrJKfa+Kox/bV3tQosyQ14V5u1s43p9CymFayJzrq8VzrqRCymlljJ2dxhtxEOxkFJaDJJZnn0OU9+83sVCi8KWkG7d585T5QFavEO1hL7Gzdwn831IKQP2BgNNJBHQ6OF6DwspBV77mKo0OhMr9kOxECmyFsjtfM0BNevpxR6klAgloGLMFG27qR6ihMSk0VE0JpNr3M+wkFJKyZJnmpnSMOb2erQLKaUOSaAnKdziPCvFLqQU+KSSaEhu82Bi3EOxkFJ4iUeX+ShhbgpXvQxCAs9vyb0iqVReUn9sSxdSCnfVOH9zS65dYRUuRMpkuDFDjDIM1rnPqnnAt48SeH1qoWaVRL8z7lx2PkRKlatNMSHx9vk2jw/kQ6SwrQsP28OT4k+NtikeBEoMXN5ioEUMzJcFT/6pmAuRIrovRYYRt9j2/RoewOK0BplG/zU5dPJOIe4iSoDOYFJdufCuh2ducxElYAbjTuWQO2XGo7JcRAmDAQiN4sgIkotgEBJ4vVrg4TsuH3WYJLtKDqIE5j6EHTgkUD49C50uooTMTjBon6Pp6hOd4CJKkE0bMei5UziuJxhECbCcSxUxI9yewB4XUQL3LOW5oTAzwSn/Dqzi1L6ZLslMck3DKf+OKIH38CZZY+uTo+wQDKIE3rCBDklxiDvO1UWUwJSgXAJj+E6/ir/JWbjQIOClWFzyo8bbxR1ECTEw4agsEca8x5oDrOIj0GiHuLS+IzVdRAkpSfJ+ims+7WmBVVyu9BMpt8PSVf4dUULOlRtZIi6ptUMwyCJgpp36WHz5u/wboMVLl4vzKO5nU9RFlAALiQFmbLgo8TKbYBAlMFuhNhw3Wc84dhAldF4kKmIYY9vddRElMAtMkZbodWft9AAtPoPsQ/CW6cZzUKe8QRahl7KGBnrk/ZENQgLT4HETRsZq3I6CB2hxWJyV2qbSQNg7sy6iBCZ7UDFtvPMGDqIEuOFST56DamFcNTKIEgpvVhMxww5vgkGUwO2BJCo2M9PRRTCIEngKBX8wAULg3aSHYBAlgMwE9zxp3ns5qs5BSIBZD1ugykRULjfSRZQAYw5KhFMdNNvO++8iSsCIgvnB1ZTGZchxCAYhgScwE7Oz8YayKmceFsFBSKCVxTAL2XWpzDuz6+RBoDA4vkHxj1dkart5jmv96vWXV3r911d4/fGFufz119fPf/n9f/zhd9//+rtfvH73wwf3nGFqMgDuk8rn/J1h9OBL7JH183+Or9//gB/6K/733Udh6FwRM0+Zl+Ar9c+gJqHK84XMI9STt95d9f3yE+/v8f70/QPXs/5mHX718f8B73HRYgplbmRzdHJlYW0KZW5kb2JqCjExIDAgb2JqCjEzNjk0CmVuZG9iagoxNiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDMyNyA+PgpzdHJlYW0KeJw9kkuOxDAIRPc5BRdoyfz8OU9Gs+q5/7YfdnoWUaEARVG4a0qT6PJSlxwq2Zv86BVTxc3k74mGxDRxN9DBKfe1owj+BLgOZu4M0UqJQb11icrykbHz19fTlwfvy9thsFF6lphXn5KxBgvydHTxAebc2u5/le8rk44svUpPRe9LvViWqA8x+2L1KdzG5hvbejCeTPFrNlGrmVPW2iK6DNRFSjex1QtQ7VGB41xHbxVtAxCBO0EHlQm1amzBSEi0RdYvzGlC98T/7QG6X9rEdeyD2LSNZ3xFGkPQ3bhHWcPMMep8anPD2tPZJuuU4kwu17I4Vgk0SstYakvwrHpdFGODGWilEM3RamxlaHbo62V4P8gMbzuKsailSs+eN6fgXH3s1xR68DyLHcGbnNgt6OeQrEIGLTZh6ixdr/A55339fgAqPX1GCmVuZHN0cmVhbQplbmRvYmoKMTcgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAzMjUgPj4Kc3RyZWFtCnicNZK5cUMxEEPzX8U24BnuxaMeeRzZ/ad+oOxAA4jkggA+O9yGuduHp/UY1nPYpz+XptvPUzutatr3U32selslKxMcy2q1vZ7cbsUvu8C2jHnx9cTal0VxYi0L58R0dnwzXW2+WIm8mEczl820GCjltpgHRb9qaenHAqV0dta5iIOxLcfBZeIsTuBO+9IGe3KmCoaeMgRxhb5DO2LB2WyT9Cao7suiFAvOqJ5AUSiH5zIvofpb9iHrJB3G4lalMWT5QBNfhCbdKUPN6QdrCHHxpIMNMOAXGJpx/59la5C9DUfYaUYVdE7zjDfegcsInTlQoWZFzCUp3ESqGKlHLZiv4ATG8l2R7MeRnzPpE4sAM4KlEBDV6H8f3Nt5K3LkRGpKCj68Xo7uF8rRm6nLbua4o/uWXZqRUoDD/p/a6/n6BZgxebsKZW5kc3RyZWFtCmVuZG9iagoxOCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE3OCA+PgpzdHJlYW0KeJw9kEsSAyEIRPeeoo8gP9HzJJXV5P7bNM6YhfYTKGywpegYzsvVkF3xllYYnvg27wueiaup9YckFTL7pg4Zi2fyFCckFK8mY0DcsdXWo3Eywgy7LCZmYK6Kr0QuqFAC6lHyamaj4Gqmgr59xFaZ9RYXCEUCJixnK/6fA9PrFh0MFpjtN8uzPPYK03QszHuGeQ9FI34G5VpuYGWtg5+GHZr3tmgo/+QnexZ4tc8P3qtCoQplbmRzdHJlYW0KZW5kb2JqCjE0IDAgb2JqCjw8IC9CYXNlRm9udCAvRGVqYVZ1U2Fucy1PYmxpcXVlIC9DaGFyUHJvY3MgMTUgMCBSCi9FbmNvZGluZyA8PCAvRGlmZmVyZW5jZXMgWyAxMDEgL2UgMTE1IC9zIC90IF0gL1R5cGUgL0VuY29kaW5nID4+Ci9GaXJzdENoYXIgMCAvRm9udEJCb3ggWyAtMTAxNiAtMzUxIDE2NjAgMTA2OCBdIC9Gb250RGVzY3JpcHRvciAxMyAwIFIKL0ZvbnRNYXRyaXggWyAwLjAwMSAwIDAgMC4wMDEgMCAwIF0gL0xhc3RDaGFyIDI1NSAvTmFtZSAvRGVqYVZ1U2Fucy1PYmxpcXVlCi9TdWJ0eXBlIC9UeXBlMyAvVHlwZSAvRm9udCAvV2lkdGhzIDEyIDAgUiA+PgplbmRvYmoKMTMgMCBvYmoKPDwgL0FzY2VudCA5MjkgL0NhcEhlaWdodCAwIC9EZXNjZW50IC0yMzYgL0ZsYWdzIDk2Ci9Gb250QkJveCBbIC0xMDE2IC0zNTEgMTY2MCAxMDY4IF0gL0ZvbnROYW1lIC9EZWphVnVTYW5zLU9ibGlxdWUKL0l0YWxpY0FuZ2xlIDAgL01heFdpZHRoIDEzNTAgL1N0ZW1WIDAgL1R5cGUgL0ZvbnREZXNjcmlwdG9yIC9YSGVpZ2h0IDAgPj4KZW5kb2JqCjEyIDAgb2JqClsgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAKNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCAzMTggNDAxIDQ2MCA4MzggNjM2Cjk1MCA3ODAgMjc1IDM5MCAzOTAgNTAwIDgzOCAzMTggMzYxIDMxOCAzMzcgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNgo2MzYgNjM2IDMzNyAzMzcgODM4IDgzOCA4MzggNTMxIDEwMDAgNjg0IDY4NiA2OTggNzcwIDYzMiA1NzUgNzc1IDc1MiAyOTUKMjk1IDY1NiA1NTcgODYzIDc0OCA3ODcgNjAzIDc4NyA2OTUgNjM1IDYxMSA3MzIgNjg0IDk4OSA2ODUgNjExIDY4NSAzOTAgMzM3CjM5MCA4MzggNTAwIDUwMCA2MTMgNjM1IDU1MCA2MzUgNjE1IDM1MiA2MzUgNjM0IDI3OCAyNzggNTc5IDI3OCA5NzQgNjM0IDYxMgo2MzUgNjM1IDQxMSA1MjEgMzkyIDYzNCA1OTIgODE4IDU5MiA1OTIgNTI1IDYzNiAzMzcgNjM2IDgzOCA2MDAgNjM2IDYwMCAzMTgKMzUyIDUxOCAxMDAwIDUwMCA1MDAgNTAwIDEzNTAgNjM1IDQwMCAxMDcwIDYwMCA2ODUgNjAwIDYwMCAzMTggMzE4IDUxOCA1MTgKNTkwIDUwMCAxMDAwIDUwMCAxMDAwIDUyMSA0MDAgMTAyOCA2MDAgNTI1IDYxMSAzMTggNDAxIDYzNiA2MzYgNjM2IDYzNiAzMzcKNTAwIDUwMCAxMDAwIDQ3MSA2MTcgODM4IDM2MSAxMDAwIDUwMCA1MDAgODM4IDQwMSA0MDEgNTAwIDYzNiA2MzYgMzE4IDUwMAo0MDEgNDcxIDYxNyA5NjkgOTY5IDk2OSA1MzEgNjg0IDY4NCA2ODQgNjg0IDY4NCA2ODQgOTc0IDY5OCA2MzIgNjMyIDYzMiA2MzIKMjk1IDI5NSAyOTUgMjk1IDc3NSA3NDggNzg3IDc4NyA3ODcgNzg3IDc4NyA4MzggNzg3IDczMiA3MzIgNzMyIDczMiA2MTEgNjA4CjYzMCA2MTMgNjEzIDYxMyA2MTMgNjEzIDYxMyA5OTUgNTUwIDYxNSA2MTUgNjE1IDYxNSAyNzggMjc4IDI3OCAyNzggNjEyIDYzNAo2MTIgNjEyIDYxMiA2MTIgNjEyIDgzOCA2MTIgNjM0IDYzNCA2MzQgNjM0IDU5MiA2MzUgNTkyIF0KZW5kb2JqCjE1IDAgb2JqCjw8IC9lIDE2IDAgUiAvcyAxNyAwIFIgL3QgMTggMCBSID4+CmVuZG9iagoyMyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDQ5ID4+CnN0cmVhbQp4nDOyNFUwULC0ABKGluYK5kaWCimGXEA+iJXLBRPLAbMMgDRYaQ5MRQ5XGgClRAzkCmVuZHN0cmVhbQplbmRvYmoKMjQgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyNzkgPj4Kc3RyZWFtCnicPVI7ksUgDOtzCl+AGfAXzpOdrbL3b1eCvFdkrERGkh3SUroMDWkjVWIMqezyMy7CtlT+DiL7XD6/0NY60JyHo4uNCaAp96WrA3YVrZIWvusoMgTmENhommhOUXAKCbV92nCQh/oA6zIidr0vBiV6NrKtQ/Yg9hOpGVJBOfCOhFCEh82+Y/g4cTztzRNgTp4MOCN79TNOKV05YTkUMDK3wy2w3tdnX89GuYi85gdhDiLHInJiUxMMkkCxHF6OdPDUU6M2A+RRXDBRx2yxwM+9Zz7oMaRT6JHt2FwupF5k8IX52aHwYD+3AAYqqrWdBjtYvb+OY72GDb+P4VrkjtuKLRygTRcOxDvBEVnv7y15rt9/B3JuBgplbmRzdHJlYW0KZW5kb2JqCjI1IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjc2ID4+CnN0cmVhbQp4nD1SS47oMAjb5xRcIFL4Jj1PR7Pqu//22bSaFRZgY0jKS5aopUwtk1SVXUt+dHTuMvk3NK9Gz7D15sxD5kmxODK3yj0M5JlbrNBrb9SrCwCxSO7UEtslxpZLxZJkh6ztEHdk0OHFKtnM+NrCDjdrjkdSFypepwcF+Yww8E7Mfb6JZSE0Vxltt8BBDxaoAz0wCxO5IuPdaxM9jXgLoKwPmb+ILjfUXI9sTIBTuNoL3qmU1jEPFYmiHOxGGr2fo+Y4ocPbPWI59nIJRwbe4mjHe/BFiJ5GVkSsfsizEfmGDalIj5jKG+puH3r8ja3YaH92JknB11/W20w10rng1Fyt1l+DR2jw3qjh8/dLnvH7HyEmbhIKZW5kc3RyZWFtCmVuZG9iagoyNiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDI0NyA+PgpzdHJlYW0KeJxNUbttRDEM698UXOAA62t5ngtSXfZvQ8kIkMIgoS8ppyUW9sZLDOEHWw++5JFVQ38ePzHsMyw9yeTUP+a5yVQUvhWqm5hQF2Lh/WgEvBZ0LyIrygffj2UMc8734KMQl2AmNGCsb0kmF9W8M2TCiaGOw0GbVBh3TRQsrhXNM8jtVjeyOrMgbHglE+LGAEQE2ReQzWCjjLGVkMVyHqgKkgVaYNfpG1GLgiuU1gl0otbEuszgq+f2djdDL/LgqLp4fQzrS7DC6KV7LHyuQh/M9Ew7d0kjvfCmExFmDwVSmZ2RlTo9Yn23QP+fZSv4+8nP8/0LFShcKgplbmRzdHJlYW0KZW5kb2JqCjI3IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggOTAgPj4Kc3RyZWFtCnicTY1BEsAgCAPvvCJPUETQ/3R60v9fq9QOvcBOAokWRYL0NWpLMO64MhVrUCmYlJfAVTBcC9ruosr+MklMnYbTe7cDg7LxcYPSSfv2cXoAq/16Bt0P0hwiWAplbmRzdHJlYW0KZW5kb2JqCjI4IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjU1ID4+CnN0cmVhbQp4nEWRS5IDIAhE956CI4D85DyZmlVy/+00mEw2dpeo/YRKI6YSLOcUeTD9yPLNZLbptRyrnY0CiiIUzOQq9FiB1Z0p4sy1RLX1sTJy3Okdg+IN566cVLK4UcY6qjoVOKbnyvqq7vy4LMq+I4cyBWzWOQ42cOW2YYwTo81Wd4f7RJCnk6mj4naQbPiDk8a+ytUVuE42++olGAeCfqEJTPJNoHWGQOPmKXpyCfbxcbvzQLC3vAmkbAjkyBCMDkG7Tq5/cev83v86w53n2gxXjnfxO0xru+MvMcmKuYBF7hTU8z0XresMHe/JmWNy031D51ywy91Bps/8H+v3D1CKZogKZW5kc3RyZWFtCmVuZG9iagoyOSAwIG9iago8PCAvQkJveCBbIC0xMDIxIC00NjMgMTc5NCAxMjMzIF0gL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAzNwovU3VidHlwZSAvRm9ybSAvVHlwZSAvWE9iamVjdCA+PgpzdHJlYW0KeJzjMjQwUzA2NVXI5TI3NgKzcsAsI3MjIAski2BBZNMAAV8KCgplbmRzdHJlYW0KZW5kb2JqCjMwIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggODAgPj4Kc3RyZWFtCnicRYy7DcA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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
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   ],
   "source": [
    "y = x.subs({omega: 10, sigma: -.1})\n",
    "\n",
    "sym.plot(sym.re(y), (t, 0, 2*sym.pi), ylabel=r'Re{$e^{st}$}')\n",
    "sym.plot(sym.im(y), (t, 0, 2*sym.pi), ylabel=r'Im{$e^{st}$}');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Try other values for `omega` and `sigma` to create signals with increasing/constant/decreasing amplitudes and different angular frequencies."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dirac Impulse\n",
    "\n",
    "The Dirac impulse is one of the most important signals in the theory of signals and systems. It is used for the characterization of LTI systems and the modeling of impulse-like signals. The Dirac impulse is defined by way of the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) which is not a function in the conventional sense. It is a generalized function or *distribution*. The Dirac impulse is denoted as $\\delta(t)$. The Dirac delta function is defined by its effect on other functions. A rigorous treatment is beyond the scope of this course material. Please refer to the literature for a detailed discussion of the mathematical foundations of the Dirac delta distribution. Fortunately it is suitable to consider only certain properties for its application in signal processing. The most relevant ones are\n",
    "\n",
    "1. **Sifting property**\n",
    "    \\begin{equation}\n",
    "    \\int_{-\\infty}^{\\infty} \\delta(t) \\cdot x(t) = x(0)\n",
    "    \\end{equation}\n",
    "    where $x(t)$ needs to be differentiable at $t=0$. The sifting property implies $\\int_{-\\infty}^{\\infty} \\delta(t) = 1$.\n",
    "    \n",
    "2. **Multiplication**\n",
    "    \\begin{equation}\n",
    "    x(t) \\cdot \\delta(t) = x(0) \\cdot \\delta(t)\n",
    "    \\end{equation}\n",
    "    where $x(t)$ needs to be differentiable at $t=0$.\n",
    "    \n",
    "3. **Linearity**\n",
    "    \\begin{equation}\n",
    "    a \\cdot \\delta(t) + b \\cdot \\delta(t) = (a+b) \\cdot \\delta(t)\n",
    "    \\end{equation}\n",
    "    \n",
    "4. **Scaling**\n",
    "    \\begin{equation}\n",
    "    \\delta(a t) = \\frac{1}{|a|} \\delta(t)\n",
    "    \\end{equation}\n",
    "    where $a \\in \\mathbb{R} \\setminus 0$. This implies that the Dirac impulse is a function with even symmetry.\n",
    "    \n",
    "5. **Derivation**\n",
    "    \\begin{equation}\n",
    "    \\int_{-\\infty}^{\\infty} \\frac{d \\delta(t)}{dt} \\cdot x(t) \\; dt = - \\frac{d x(t)}{dt} \\bigg\\vert_{t = 0}\n",
    "    \\end{equation}\n",
    "\n",
    "6. **Convolution**\n",
    "    \n",
    "    Generalization of the sifting property yields\n",
    "    \\begin{equation}\n",
    "    \\int_{-\\infty}^{\\infty} \\delta(\\tau) \\cdot x(t - \\tau)  \\, d\\tau = x(t)\n",
    "    \\end{equation}\n",
    "    \n",
    "    This operation is known as [convolution](https://en.wikipedia.org/wiki/Convolution) and will be introduced later in more detail. It may be concluded already here that the Dirac delta function constitutes the neutral element of the convolution.\n",
    "\n",
    "It is important to note that the product $\\delta(t) \\cdot \\delta(t)$ of two Dirac impulses is not defined."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "This example illustrates some of the basic properties of the Dirac impulse. Let's first define a Dirac impulse by way of the Dirac delta function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\delta\\left(t\\right)$"
      ],
      "text/plain": [
       "δ(t)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "delta = sym.DiracDelta(t)\n",
    "delta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's check the sifting property by defining an arbitrary signal (function) $f(t)$ and integrating over its product with the Delta impulse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(0 \\right)}$"
      ],
      "text/plain": [
       "f(0)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sym.Function('f')(t)\n",
    "sym.integrate(delta*f, (t, -sym.oo, sym.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Derive the sifting property for a shifted Dirac impulse $\\delta(t-\\tau)$ and check your results by modifying above example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Heaviside Signal\n",
    "\n",
    "The Heaviside signal is defined by the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function)\n",
    "\n",
    "\\begin{equation}\n",
    "\\epsilon(t) = \\begin{cases} 0 & t<0 \\\\  \\frac{1}{2} & t=0 \\\\ 1 & t > 0 \\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "Note that alternative definitions exist, which differ with respect to the value of $\\epsilon(t)$ at $t=0$. The Heaviside signal may be used to represent a signal that switches on at a specified time and stays switched on indefinitely. The Heaviside signal can be related to the Dirac impulse by\n",
    "\n",
    "\\begin{equation}\n",
    "\\epsilon(t) = \\int_{-\\infty}^{t} \\delta(\\tau) \\; d\\tau\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "In the following, a Heaviside signal $\\epsilon(t)$ is defined and plotted. Note that `Sympy` denotes the Heaviside function by $\\theta(t)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "θ(t)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step = sym.Heaviside(t)\n",
    "step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(step, (t, -2, 2), ylim=[-0.2, 1.2], ylabel=r'$\\epsilon(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's construct a harmonic signal $\\cos(\\omega t)$ with $\\omega=2$ which is switched on at $t=0$. Considering the definition of the Heaviside function, the desired signal is given as \n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = \\cos(\\omega t) \\cdot \\epsilon(t)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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SgRAStC6GgUAI+OcUCTaHttIKkWPpORvSzZigc92E3Y1DOS3ARTEhUXRuPk5RCAIMpd92byEQSGoxAsziqGRMPS/pbgGqzhp1LWVlnjG3fWfCAlQdIdmLi+3I8uUMxnoBCxFBi6CzdA5Eqy6/bTO0EBGqTkN1QKtf1a8bExag6kSXiNycBxZkqiu1MRERUIW9dAaLAZEYc3uADGQSWggSCEk37tLeRzMR3egkg8FRIPJ0IurIQBfBQkTw6mnTTTKd3ZMfrjGykElAwWP50v+k/2XFZxMRgYSksmhHpoovC2sHxUREGFuJkqcj5UFrb4KBTEKr1c8DSMY+tXgRbogI5F6oMz/ve+oWxn5pA5mENq6Xzkwy7P1fExFBx0Xz2g+JYfR1HSmZyCQg+Ssps5JnTU+9CDfkbdzIRQE+e1qzz223YCCToB2Dmegxn623i3BDRKisJRzJyCRxouUiGIgIOPdcxz0hdCbtLglqIpNAKNdp1ryHkC7TMJB5YZnYF/vYiEFh96tLBgKhaMsp6lQ04eTadSJrIiJMSY5s0mZSR5Yun2chIkjCJeV3OkOuwbv10hYiAn686RqIUkNebO1AW4Cq6+ZH1bGPjgpLvarfAVXX5Efp0ZI1SdmvtWAhIlQJKt2c0AFkb9tHGgDVyR2LrqQkef7qWtn9sRARtEmO42mHJ5QzKHltetmQKO0spUjo+7FfktKeBBOC0nAgMejrh7FfXlpeG182pNuf8ps9avc+6sb5qDIpJiQKOtRXbdmHkaVruhbFgkShSZ8V+fDpp6TyOjW1IVG0rTq29pTiIYD8dk8mJIryHG3Aj9375xWaJ8WCROnaT9Z1XImk6nRcsigW9PZIIyOqXce8ujjZWG7rFqwJiaKLW8j9eZMuO1fXdqENTUrJrrU40jyXal2Zkw2JEnSDIY/TVAW3mlceYEOTIluN8w5hrtcVGxsShegTnPLokRuWvI7MTWQS1Fycp1BRWbu/GHdIFGKCZyWEIfUdZhT2y1vQpBBpfZvHYxmpX/NFuUOi5BPTyc9zTofprYTOhiYl9azNPpUTLXZEs6FJIXC5Pk8gdSgeX1q5Q6Kg93TW9cwmma3diIFMgtLBFtZ93FTixbhDokjL6QLL8/g1Smg+KRY0KbxgWoe83r0YvgWJ0pSy1vK8ZMvMrRVpQ6JoK0FXlpTk9VLcfpM7MKs32cNM8VpBefiLcIegkBiF1EOZ5UHfQq3xNSFR/HCFfR7uk4bFvYBNaFKIRrp6rSWXSBP2vJuQKEFXSN3wBbrziI9aZm9CosSRete551R8zssX29CkNMSQ8sKukUx719yGRNGVzSIBqasdiM+922lDouBlvQ74h8PNiNS8X9+CRJEoirqlM66pqPN7kC1IlIZnqmPwa9CxvW4SPykWpC+wdBKOpCV4IOAaS3s7ChN699EWqlvXY9symG+On49w/NP8zHJ8yPj+84zrw5j5ueTLhzL7o0gesz+tHB9Cvvuo8s34qPLx0/6Icn08OT60fP1Q40l6/13Ah3ZeP878O+188/gf3XFnbQplbmRzdHJlYW0KZW5kb2JqCjExIDAgb2JqCjQ0NTUKZW5kb2JqCjE2IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTc4ID4+CnN0cmVhbQp4nD2QSxIDIQhE956ijyA/0fMkldXk/ts0zpiF9hMobLCl6BjOy9WQXfGWVhie+DbvC56Jq6n1hyQVMvumDhmLZ/IUJyQUryZjQNyx1dajcTLCDLssJmZgroqvRC6oUALqUfJqZqPgaqaCvn3EVpn1FhcIRQImLGcr/p8D0+sWHQwWmO03y7M89grTdCzMe4Z5D0UjfgblWm5gZa2Dn4Ydmve2aCj/5Cd7Fni1zw/eq0KhCmVuZHN0cmVhbQplbmRvYmoKMTcgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA5MiA+PgpzdHJlYW0KeJw9jLENwDAIBHum+AUiYYxt2CdK5ezf5i0naeD0D9fSoDiscXZVNB84i3x4S/WEjcSUppVHU5zd2hYOK4MUu9gWFl5hEaTyapjxeVPVwJJSlOXN+n93PcerG7oKZW5kc3RyZWFtCmVuZG9iagoxNCAwIG9iago8PCAvQmFzZUZvbnQgL0RlamFWdVNhbnMtT2JsaXF1ZSAvQ2hhclByb2NzIDE1IDAgUgovRW5jb2RpbmcgPDwgL0RpZmZlcmVuY2VzIFsgMTE2IC90IDEyMCAveCBdIC9UeXBlIC9FbmNvZGluZyA+PiAvRmlyc3RDaGFyIDAKL0ZvbnRCQm94IFsgLTEwMTYgLTM1MSAxNjYwIDEwNjggXSAvRm9udERlc2NyaXB0b3IgMTMgMCBSCi9Gb250TWF0cml4IFsgMC4wMDEgMCAwIDAuMDAxIDAgMCBdIC9MYXN0Q2hhciAyNTUgL05hbWUgL0RlamFWdVNhbnMtT2JsaXF1ZQovU3VidHlwZSAvVHlwZTMgL1R5cGUgL0ZvbnQgL1dpZHRocyAxMiAwIFIgPj4KZW5kb2JqCjEzIDAgb2JqCjw8IC9Bc2NlbnQgOTI5IC9DYXBIZWlnaHQgMCAvRGVzY2VudCAtMjM2IC9GbGFncyA5NgovRm9udEJCb3ggWyAtMTAxNiAtMzUxIDE2NjAgMTA2OCBdIC9Gb250TmFtZSAvRGVqYVZ1U2Fucy1PYmxpcXVlCi9JdGFsaWNBbmdsZSAwIC9NYXhXaWR0aCAxMzUwIC9TdGVtViAwIC9UeXBlIC9Gb250RGVzY3JpcHRvciAvWEhlaWdodCAwID4+CmVuZG9iagoxMiAwIG9iagpbIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwCjYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgMzE4IDQwMSA0NjAgODM4IDYzNgo5NTAgNzgwIDI3NSAzOTAgMzkwIDUwMCA4MzggMzE4IDM2MSAzMTggMzM3IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYKNjM2IDYzNiAzMzcgMzM3IDgzOCA4MzggODM4IDUzMSAxMDAwIDY4NCA2ODYgNjk4IDc3MCA2MzIgNTc1IDc3NSA3NTIgMjk1CjI5NSA2NTYgNTU3IDg2MyA3NDggNzg3IDYwMyA3ODcgNjk1IDYzNSA2MTEgNzMyIDY4NCA5ODkgNjg1IDYxMSA2ODUgMzkwIDMzNwozOTAgODM4IDUwMCA1MDAgNjEzIDYzNSA1NTAgNjM1IDYxNSAzNTIgNjM1IDYzNCAyNzggMjc4IDU3OSAyNzggOTc0IDYzNCA2MTIKNjM1IDYzNSA0MTEgNTIxIDM5MiA2MzQgNTkyIDgxOCA1OTIgNTkyIDUyNSA2MzYgMzM3IDYzNiA4MzggNjAwIDYzNiA2MDAgMzE4CjM1MiA1MTggMTAwMCA1MDAgNTAwIDUwMCAxMzUwIDYzNSA0MDAgMTA3MCA2MDAgNjg1IDYwMCA2MDAgMzE4IDMxOCA1MTggNTE4CjU5MCA1MDAgMTAwMCA1MDAgMTAwMCA1MjEgNDAwIDEwMjggNjAwIDUyNSA2MTEgMzE4IDQwMSA2MzYgNjM2IDYzNiA2MzYgMzM3CjUwMCA1MDAgMTAwMCA0NzEgNjE3IDgzOCAzNjEgMTAwMCA1MDAgNTAwIDgzOCA0MDEgNDAxIDUwMCA2MzYgNjM2IDMxOCA1MDAKNDAxIDQ3MSA2MTcgOTY5IDk2OSA5NjkgNTMxIDY4NCA2ODQgNjg0IDY4NCA2ODQgNjg0IDk3NCA2OTggNjMyIDYzMiA2MzIgNjMyCjI5NSAyOTUgMjk1IDI5NSA3NzUgNzQ4IDc4NyA3ODcgNzg3IDc4NyA3ODcgODM4IDc4NyA3MzIgNzMyIDczMiA3MzIgNjExIDYwOAo2MzAgNjEzIDYxMyA2MTMgNjEzIDYxMyA2MTMgOTk1IDU1MCA2MTUgNjE1IDYxNSA2MTUgMjc4IDI3OCAyNzggMjc4IDYxMiA2MzQKNjEyIDYxMiA2MTIgNjEyIDYxMiA4MzggNjEyIDYzNCA2MzQgNjM0IDYzNCA1OTIgNjM1IDU5MiBdCmVuZG9iagoxNSAwIG9iago8PCAvdCAxNiAwIFIgL3ggMTcgMCBSID4+CmVuZG9iagoyMiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM5MiA+PgpzdHJlYW0KeJw9UktuBTEI288puECl8E1ynqne7t1/W5vMVKoKLwO2MZSXDKklP+qSiDNMfvVyXeJR8r1samfmIe4uNqb4WHJfuobYctGaYrFPHMkvyLRUWKFW3aND8YUoEw8ALeCBBeG+HP/xF6jB17CFcsN7ZAJgStRuQMZD0RlIWUERYfuRFeikUK9s4e8oIFfUrIWhdGKIDZYAKb6rDYmYqNmgh4SVkqod0vGMpPBbwV2JYVBbW9sEeGbQENnekY0RM+3RGXFZEWs/PemjUTK1URkPTWd88d0yUvPRFeik0sjdykNnz0InYCTmSZjncCPhnttBCzH0ca+WT2z3mClWkfAFO8oBA7393pKNz3vgLIxc2+xMJ/DRaaccE62+HmL9gz9sS5tcxyuHRRSovCgIftdBE3F8WMX3ZKNEd7QB1iMT1WglEAwSws7tMPJ4xnnZ3hW05vREaKNEHtSOET0ossXlnBWwp/yszbEcng8me2+0j5TMzKiEFdR2eqi2z2Md1Hee+/r8AS4AoRkKZW5kc3RyZWFtCmVuZG9iagoyMyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDI0NyA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   ],
   "source": [
    "x = sym.cos(omega*t) * sym.Heaviside(t)\n",
    "sym.plot(x.subs(omega, 2), (t, -2, 10), ylim=[-1.2, 1.2], ylabel=r'$x(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rectangular Signal\n",
    "\n",
    "The rectangular signal is defined by the [rectangular function](https://en.wikipedia.org/wiki/Rectangular_function)\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{rect}(t) = \\begin{cases} 1 & |t| < \\frac{1}{2} \\\\ \\frac{1}{2} & |t| = \\frac{1}{2} \\\\ 0 & |t| > \\frac{1}{2} \\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "Its time limits and amplitude are chosen such that the area under the function is $1$.\n",
    "\n",
    "Note that alternative definitions exist, which differ with respect to the value of $\\text{rect}(t)$ at $t = \\pm \\frac{1}{2}$. The rectangular signal is used to represent a signal which has finite duration, respectively is switched on for a limited period of time. The rectangular signal can be related to the Heaviside signal by\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{rect}(t) = \\epsilon \\left(t + \\frac{1}{2} \\right) - \\epsilon \\left(t - \\frac{1}{2} \\right)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The Heaviside function is used to define a rectangular function in `Sympy`. This function is then used as rectangular signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "class rect(sym.Function):\n",
    "\n",
    "    @classmethod\n",
    "    def eval(cls, arg):\n",
    "        return sym.Heaviside(arg + sym.S.Half) - sym.Heaviside(arg - sym.S.Half)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(rect(t), (t, -1, 1), ylim=[-0.2, 1.2], ylabel=r'rect$(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Use $\\text{rect}(t)$ to construct a harmonic signal $\\cos(\\omega t)$ with $\\omega=2$ which is switched on at $t=-\\frac{1}{2}$ and switched off at $t=+\\frac{1}{2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sign Signal\n",
    "\n",
    "The sign signal is defined by the [sign/signum function](https://en.wikipedia.org/wiki/Sign_function) which evaluates the sign of its argument\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{sgn}(t) = \\begin{cases} 1 & t>0 \\\\ 0 & t=0 \\\\ -1 & t < 0  \\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "The sign signal is useful to represent the absolute value of a real-valued signal $x(t) \\in \\mathbb{R}$ by a multiplication\n",
    "\n",
    "\\begin{equation}\n",
    "|x(t)| = x(t) \\cdot \\text{sgn}(x(t))\n",
    "\\end{equation}\n",
    "\n",
    "It is related to the Heaviside signal by\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{sgn}(t) = 2 \\cdot \\epsilon(t) - 1\n",
    "\\end{equation}\n",
    "\n",
    "when following above definition with $\\epsilon(0)=\\frac{1}{2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following example illustrates the sign signal $\\text{sgn}(t)$. Note that the sign function is represented as $\\text{sign}(t)$ in `Sympy`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\operatorname{sign}{\\left(t \\right)}$"
      ],
      "text/plain": [
       "sign(t)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgn = sym.sign(t)\n",
    "sgn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "sym.plot(sgn, (t, -2, 2), ylim=[-1.2, 1.2], ylabel=r'sgn$(t)$');"
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   "source": [
    "**Exercise**\n",
    "\n",
    "* Check the values of $\\text{sgn}(t)$ for $t \\to 0^-$, $t = 0$ and $t \\to 0^+$ as implemented in `SymPy`. Do they conform to above definition?"
   ]
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*."
   ]
  }
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