{
 "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": [
    "## Elementary Operations\n",
    "\n",
    "Operations like superposition, temporal shifting and scaling are used to construct signals with a more complex structure than the previously introduced [standard signals](standard_signals.ipynb). A set of elementary operations are introduced that are frequently used in signal processing."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Superposition\n",
    "\n",
    "The weighted superposition $x(t)$ of two signals $x_\\text{A}(t)$ and $x_\\text{B}(t)$ is given as\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = A \\cdot x_\\text{A}(t) + B \\cdot x_\\text{B}(t)\n",
    "\\end{equation}\n",
    "\n",
    "with the complex weights $A, B \\in \\mathbb{C}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following example illustrates the superposition of two harmonic signals $x_\\text{A}(t) = A \\cdot \\cos(\\omega_\\text{A} t)$ and  $x_\\text{B}(t) = B \\cdot \\cos(\\omega_\\text{B} t)$ with weights $A$, $B$ and angular frequencies $\\omega_\\text{A}$ and $\\omega_\\text{B}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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qRIwAm0ndOAwEgNk02mHckFFmz2xr7bS+luRDuSFSIMsoCKeWJxhc22cTQ0aBVtuGbblhYHyEmUSQUSCh6CnWbUBol5+UGzIKlBF661leuOHSDuWGjII1aXXbFodR6iZ/DJFCgQsRYhumCxN1WzgxZBTu6NNbxj1WrGm5HsoNGaVhfS227ztWcikZIUaAwNgbxRUCqx3ChRhBSoIA2JvXzdeUECKFlkdf0p8NcpHT8xFklFLaeh5XKN0/KDdkFGiVU/dnudGPi7VDuSGjDJkWcIEHwRra16HckFFkMKjDRkTL8vlgN0QK3RaF3maWQ91yFTdCjNA4gqxYqNXUw7gho0wa1RaaWqmX5EO5IVKG7q4NEwaqB/qAjCCjdKi+7dl05p7JB+WGjCKTcTkqpNCgJ0gphEiBNQvLblmYjXRp3TsygowySrExQduk9iqHckOkyLuggwvDhQoDkror7CFECqypCjnbdR2YUntyURxBRpllYunT1aYzzrIfyg0xMo1u4M5wAca3JpiHHh4YQqSgk1qlLsoA3c6QMo9miyBSGJuU6ZHOg1p59ckSIUYQGG2M7GOIGMTitvtjiBQeH8hQNzQMLXHDyR8rgoyynpjvAZUDq8d2A8YQKdD5KnVZ3VOA7FzbNo8howjn59AIvAEN0xfuEGJwHuxXhrtbWGCFYra9NjFklF6h0lUNPYSK1n1OhpBRGEnbrBxjde8NhAgJsPpqaxZuiWdYbhBFiBE6tP1uAZSVAQX1MG7IKDM/EaPQSD1CJwKsOgOGs0V0joRp1w/hhkjh+l9K31GpKPB7RJBR0DeYqhb5Sk15HMoNGaWlobOH5ZOq36HckFGwBHI7wwJyoT70Q7kho0Co9frE8GJany6JIKNAF+vT+orxYPJBuSFSuKGcNdBOw3KHm6gRYgRoPKqVsbc+3HYxZBTqS+sZDxAc44NyQ0bBa9FRyPLMnbiP57ohoyzua9rAhhwYLu1DiBQYfPWZO4PurT3fA8Cq15bbtBDg2jGa5BBuyCiDZli3IODUPOQnhowCs3vKIwU6BvcH5YZIgYG4IPbFZE2DCuiUCDIKbjjFQoHROu6TiyFSBheNVR+5iSf2hS6EjDJGZtAEDyEUmLouuEOIlMk4i85mGRiBTTzEM4ZIEYZOJGqYHVIXevKZjxFECu37SYMiM2ZjDA/zjhCGSDeNwnvssIIJu93bMUTKVP2Sbmwa1Awj3sMlhEiR98BKC+1Nj8dhmdmjPkJIWFTBuTUtWP3Q075eRwgjt2EcMopbD97UVbpb5hFiBJka0yJ0VLXlAiJCSOB6gWH64qYF1ny3TgLAqnfuhbfXRAf15jv2IWIEoaI/X1O4iSq+SkcICeWdIYiXFieMTZ9PEWIEGFtU27gJWKG19UO4ECNAJNGu4W4nVGnvtgghgVsCtM9YjHF9dNkIMQLW9g725H7xmu6FiBAjQO0uvHGH+UXF9RAuxAhQUqAUceN75uN6CwCrjrfH+sBiqAc/uP6FGAGvxXA8FBdo+OODcCEk0FoqVBq5c48F3o3pCDECJPvATJrcPE6+eRMiRoAwzezNTvdoHqf+jwGr3gdWHn2xBHXWxV+EGGFiHVhaCin6ef0fA1YdmlmH1cti4XJ96l8ICZ3xva12a7h61JAIMQJdzUOvU7EgunsmQowAW5/nmVDcUzniJUKMMKBR2qthniyXqxFiBEkDjc3iVZfHEIXItx5mS5CzRSdIQau73ytCjJAXFoHFYqxR6wyLADFCL+iWrNuhnLzrEC7ECAurAJQdbqfjxyMvAoSEiTaGfO2UbJyBrkdEiBHGyjzeA9mJlXmcoRQgJAgdAoWHMqnEruJ2SYQYobXK81xSKWy7qwMRQoKFvg1dXxg0cvohQIwwZ6WrbPFJe3EdJUJAmLRtJuT4a9GbLfWcGQoQEtT7R8lGZzYPFbgzLYRI0fDIPMwPkKgpOCWCSBl0R3MfjSclKUBdMIUQKYv+4YWxTw82KpxRHkKgoK0zo88sKhFCLvlBtxAiBVKI7c79MUZvL4+Pj6FvPT5a06JxV9QPejSUCDECFhu8IBRcqAvrWNUR8q2nu6HT8/0K3TQJs23PphAyCtYnDJ/CY0kYez4UI8QIUG8abljoPtFAk8O4IVIwzyBbuPHMUzdQMtztHEJG6bDJ2ewo72ic0yMRZJTJwIei5ZBHxXXZEDIK5Io2SmWMY82fD3ZDpFB36VoIfWn59t9dbpUpuLhjhXLMnOmGaAgZpcFCZXtgFtV+dOUIMcKgK2JpeYZm6IZVCBll9jlX1nKoDO3jLS7ECKsmxvCwGGplcZ0rhL718HSaQkOb5RBp1b1uIfT9nLeGSiJWjlG9Pig3ZBTYiwyR0vLWPMg1hozSm2iMNsthPMk4lBsyCsao7rjzHTEVzpCPIKNgedDQbjYkBMjnXW7IKItxR09vpenu4wgBAfI7YxokG6Jr4qK7H0PIKFiCmja9HkJeLiBCyChoCUpwHdtrHKsyhIyCFaMsK690945DuSGjQAjI2jOUkSaHckOkZIoYPQPDo3m08PxdIsgotVd6siigMJncyRMhRhi5J45unpEsbbozMISMIrMz8FHlpuB15VBuiBQ9DjJb1fIECep+8BAyCjc6eHyGGxhQJ10ShRAptF2GcNzBcl5qUG1KBJHCfZ5Z29KVLLXpx/BiyChzVI5Q6LUMjHXlKUJIoCKJhx66eY41eflOaQiRQknYKpUFRtNRRXdKBH1/GT8N9fRj7SxN9nPFECm05BlQ+dIAEh4u32eWQ4gU3h0WzHwtxo+s5Cd4I4SERZnB03pCE5d+mU0IEBD0rJSe0BAe1IeSv+8QIUZoKzHiDhooVv25PSkhYgQZU/0GA8bSNn2DclZWLSpR5YaElnPOI0SMgHdiBD51dDzjdhmHiBEYnQeTajKQfO1duAhgda5ag6c4GDgNIVv9gQLECA3CdWhxEZhv7RAuxAhDoNqoPbwgY/zQdYQYQWaiiqO2GMOLD+FCSKg0m5cZxA3zcq82IWIE/GXYvzoBpgfKhYgRmPkkiVms2XfEQsQI3Lh5jGhZsj7ucCFGmFDJ7dWYb2h83OFCjLD6TCs/Vr0LpxAhgedWIUGnuQ2Wx26FiBFqrVwIp27gljO0A8QIPWdINvV8wIjpH4QLMQLThYzHGYM5Pg7hQoyAtbnqgOnAyzr1fwyweueLLSvmdofLsAgxQsUAy5VDmFvDZR7ChRiBQWX0rfAM7kqjHsKFGAErWmfT0eVW3ZMVIkZgngrIGh4oWHLSNEQICZh8SdQdAy2+t3V6IUCMwIgJelcYASQfQy9ASNCtTIHxL8w81VtzERAgRkCnwJoQnvusvocVAawudGsx9Eroratuj4aIEaToMUKaR6LHv51wISQsRifTt0JlSTDk/ZUDBAS6znnszVzi6MvlcjKESCl6Xl3zJTCcdHmIYQyRIlSOC/fneBh3ZYybTYkgZiJhDqHReBq9812xzux3DyGjQIMeNWtSE7pY6gflhkiBttcKA0LVWynDEyTEkFEmY6Km7qBgHsg2mmKIlML4ft0upuOo9zObQsgoGKHcaKInSNAotRzKDZHCfeKi8VHcQMJSu0+nxJBR8JCD5ZNRzt3dSTFkFGiSGBe6GTYpLT8oN0QKxDB0Yt06LBpufvolgoxCd1yxjT1GFc16KDdkFIFB9Gy1ZmltfVBuiBQI+UXxajunHLn+YBFklJZS0r3Ayljq0vOh3JBRoLA+2/JonrT3VEPECAJTuD6bxvxxHsYNkYLhNi2dh5aftBcxZBQ+YcvP3ed5kQAxQqO36NmXh0qQ22HckFGwYDOoXZsEFkj+uMkNGQX6Z34SSkF+ePBUDJFC4VxTsrtDZjXXxELIKIVHaMoTXoFnb4dyQ0Z58lwV5k0onjInAKw6BAEVOR3ZPCx/6l8ICcKNpzSf2dOWxw3FkFEYXM8kJShP8+jQEWIEYeBN10ldupsAAcDqUNjReHMf9+0eNBNDRhHKcLHUVVm9n065IaaQSlwrOxuc+yFZ1j7NE0OkMJVbmpq+im5ZjOndViFECv049MlwF1gG14/NCBASGsPEmYyCO+BMR1D8sSKIFBpookE+lfExMl3DDyFShLfM3CnX4Djp+wRiDDGDFpe/Prku00cMW9gFXQgZZWnQHZQAvKBPj7uclbMee+ZamRnm1ZKv7yFklIl1SENz6ZQdJ6lXgJBALyQsOws31Aj+TQgQIwxawVmjIEdJ6bxEBBmFuayKpfDJeNH1QbkhUqpmRymWhAhWrh8FjCGjMNTqSWjFcLPtEY0ho/Ckz5OJCF3kXp4YMookxtNbOiU0Tf+g3JBRMFuY/U6DZgWreT+UGyKlMV/maBZnK32s8/oRZJTKE5hPzDDsP1/bQ8gonX739kQmS9vhUjFklEkt6QkzLt3XhAgxAqzutmTHJU/5eKwbIoUDgSbfE5c8facshoxSpU8L4mZypvLBuBAjMPxTs241HuIYZ0hGkFGYCrOMJ4R9+DGYGCJFd2Q1zIVhya27LhAhRqi9WjIqjXXyKPIYMgoW7NKeXLr0k8qh3NC3JgTmkc9pUcmlQaf0fo8go3T1lqn0YBiLe95CyCgLElyPLwjPHZaVD+WGSBHVPPsjCMsaLrZDyCiTRzSb5spahZrFodwQKUxGuxgNYScZTpRRDFmiZZgr6l0sdFWt4k6+ECIlM8owsyUrTyANOekSI4gUpiKDQWindhqMexcUIUQKdy7qSs/JXy6yu5FDiEkQmT9u6bkdHgOW6ik+Y8goiyfOmEoqMQGPx7rGEClMw1aZRJLnfwczB+5+CSFSIKXRkskyYlWob67khJBRsKjxdAQdwAMmmxtnIUQKQxvb1ENuoqeA3QMTQkZB6+kWHI/1Yi0/DxZBRmFyqPwcvrN0wk65IaMsqHti6bemJt84lBsihakZNIxOTwXKxxQLIaNUGC/VjvbqsjAO5YaM0oT7zpYTqa7ZPyg3ZJRB78+0c5Rd3MyOECNA5g7ujvGo5srH+xFCpAzNHWWHO4tA2fB+DBAjlKmmnR5TXdRPDuOGjNIwdMpzrleqC5cIMUIXjREzZzUG+gfjhowyIW2K+LFely0hZBQOoL6P9Y7lS1EIkTKZRwpTwcp5mN8fLIKMwtFQ+tOOpSc5lBsySucUeM7o1unZZ2LIKFACSrKccBiwy22hEDLKypKLHYYdnylHQ4gUxgIs5mjWzHMQpKdfIsgoENLUe1iesXy6FyeEjII2oQTUvZkqpddDuSGjwIxkUhPdLqpYQD4e7IZIWRr1bVkJ8mwfPRkgRuhQZatl4FrzBC7HkFFW1YSMFLjQnfPp+wgCRSDMYB+x6RPT0fS+Y2NiyCgYEZ2nmhOGEVOV1UO5IVKYlS1pGm0mcoSA3yc7YogUrp2ah3olHkA72k6EkEAvI5OQYeVkOEvNvqqEECmcQzx192I4NJSTo0qHECiLufJgIrSXpr7IntAsREhQfxtUbAYoN1iJ7naPEBI0e7VuxvFU526nq5hVB3c0mUaQXgqRfK4dICTgrRjY11+M8avQyf3yAWKEJZmez8E9tnrcERFCAr+eMGm1QF/CoK6+dRMhRkAhs6B0RmdMD28JERL4RZPRs2bbYmz8jiQIESM0rteaoEeYIXMdwoUYYZbJc5Vd96dlRzeEiBEoiprmI4L5lfbhkhBhXmGGaHXNpkR9XZ1ySggRI0BwYzJp4q3mYfwRYNUxxelKpl2EptsxtiFihJn5nQPNvZW7J5QLESPALoMdqOmjMGn3yh8iJHAcdu6N0xxsySNgQ8QIDPQc0xJfjbFnWogYoXHTwNJwdc+9HgFWvfPg3Xyydo1T+0flVnlO5g/TZyy5lI+LX4gRYIUxcx9bAbbT3oYNERKYHYMnOjXBFg8k7fo3YNWZdb9ny602SvqofyFG6EzZlG2kJA8ACBEjzMGNaMsNlzwYN0SMwByL0L6Zq1F6bx93uBASeIhzTDbdYKbx5smsI8QIDKBdmgyvQVks9RAuxAiQTIzHgUSAhPLg9xAhoTEsLFWhzBkj9TPRAsQIFDRossFkuuL5vgPAqsvQbK/MoAX5vPWGECEBYwT6OTTI0XiWJnsT3YBVhywemEFDzadS5dS/EBJ0oUCfvyYPxvQz72+A1ScFIJ9yMmvS6mcUBQgJPIuVFtp5aUq4dAZFgDD/OA+rLyY80/OSDLvyFOoRRMrkeQR15GDxQp/mkxU9gkhhUCPT5Lw0PDeltJWFGGLqcgpCOjPsgzf8MMYWeSFECkQJZD8tQMr17PtjIUIC4xj13BuNbEZie3+EkFEGmiJZItWGPnZJFkKkMKBLpJpdvtBpwykRZJS2uP+tRjY/orDXxBgyCtqbX0DQGKxZ/ChMDJHS6MnYRjaT8nr2+RAySmPCQDOyhd+9aIdyQ0YZmjdSzQAeQy6HcSFGgNSV+STAhlYr+TBuyCj8qlA2g4aJdD9e5EJI6DzUpR4ZWk3MMbXHcAgZpdHF+yTWWuc1bsCqY11Laz1pnrP41w9CyCjQXTQ0kzYmdwbmodyQUUDWUGk1sZnk9lBuiBTusLaS12MvFw9RjyGjFG6L7KRaoy05lBsyClaR2Z4c2rNK/bjLDRlF89A+D0x34TqUGzLKHBo/ZLnGSz29EkFGwfThIQ+1ymkzzUO5IVK4LZtErPFTH2cxDyGjVEYY27ib7SydEWKEUU+6q8zQw8O4IaOsrIfudcLlE9IXQ9/6EbrMlGJmLevZYG+vCDIKU5pXy3eFBTadGR9BRplpqLgRTcLf82FcCAnMZ6FJi1ieoPIPH18RZBS0hAZkJ37Z68S2x5BRZHCzVeWz5m+bh3JD/FyFqu/MykgDu3LU7l4JIVIyt56afqhtaZK8beTGkH1AkPlimO2KmlvKfo8bYHU9bsgPjvAsUSrHeIgQfhIj6VeZmGODaS3WOI8UQqQwImbp8e6qctP93zFEStHDfvwSZ9Hv9ol3ewiRUqlr6y4lJftM6XysI4JIaUziq2nGGfHFJI/+QZAIMkpjBjXbFRGeL+yHckNG4UIulh1r8gDSOJQbIkVDGeXZOp7jQ18JIaP0ymhpy2KDPvNVPoSMspjCaFquHJhve88zhkhhuOG0FFFcqWrfftAYMoqqilaeGNOYD+WGjMLAqGT7urhmzXIoN2QUjDeOHs2QBbHgkiWESKGbSPKzeczTR3sGR4gR+OEw3T7lPqUams64IaN0/dTbk75p1U/KDRllLPVF6Y5r+fguTwgZRbin8+wd9+37jwBW54HMNcbzmahyUvLFkFGYyniKl/u3iELIKPyAzRrPvTU7v1NuyCh92OflWL74y6HckFG4j52ezfnJ1PaHckNGYYj9s6XNbwP5161CiBSeepx5BxpQHPiDRZBRGEIpT4Is6PxuC4WQUZh+6dnUFn7ppR7KDfF7Q4mudP02HaOZIX73EZgYMgoGdGoWACKwmVyAh5BRIGeS2B51O19+CgBWZ77M1jVeZDLI0TcYYsgo/NKXPNE10Mt9xocQKcx60vTraNxshkx0R1wIkcKTAF2j6AqDP6bv4sQQKQzLWFO/5bfwozRXPUOIFKoWiyGEjQkV9KjCwwgQfgOKnxnJ+jU/OjTKOYcfQ6Rg3ZTJs+0a6jWqf1gvhkih50edKzSnR/bDPyFCAtPjLF3PmGKFUQa7tULIKNBZc7bIWSbJ2tujMUQKQ2Voq1q4Huq5ty2EjCJJPzPGcn7IqeVDuaFv/QByrhBrT7CtTM/nFUNGaWIaAKMbsWy4fyiEjMJYH7FcTIXxJvVQbogUWkl0uGqsbfeU5xFg1TmT1xNoe3zDAWDVByTSHE8+KQiaj+vfkFFW4Ub8EzM7zmwPoW/9SHrhDuh8gmbPZ9UixAh61teibBkH7JZDCBkFChwNUIuZ9aQtIfLt33E/H9TV2r98/eFVXv/JPiT/gw+1/+hrvNd3sH/wHWS9ln9B/k/6FfnPL8f7/c+n4r9+7x+K1/ueD8+fj38/Vzjfhr5u8oPvz8d3+eXX/wWFr71xCmVuZHN0cmVhbQplbmRvYmoKMTEgMCBvYmoKOTI5OAplbmRvYmoKMTYgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxNzggPj4Kc3RyZWFtCnicPZBLEgMhCET3nqKPID/R8ySV1eT+2zTOmIX2EyhssKXoGM7L1ZBd8ZZWGJ74Nu8LnomrqfWHJBUy+6YOGYtn8hQnJBSvJmNA3LHV1qNxMsIMuywmZmCuiq9ELqhQAupR8mpmo+BqpoK+fcRWmfUWFwhFAiYsZyv+nwPT6xYdDBaY7TfLszz2CtN0LMx7hnkPRSN+BuVabmBlrYOfhh2a97ZoKP/kJ3sWeLXPD96rQqEKZW5kc3RyZWFtCmVuZG9iagoxNyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDkyID4+CnN0cmVhbQp4nD2MsQ3AMAgEe6b4BSJhjG3YJ0rl7N/mLSdp4PQP19KgOKxxdlU0HziLfHhL9YSNxJSmlUdTnN3aFg4rgxS72BYWXmERpPJqmPF5U9XAklKU5c36f3c9x6sbugplbmRzdHJlYW0KZW5kb2JqCjE0IDAgb2JqCjw8IC9CYXNlRm9udCAvRGVqYVZ1U2Fucy1PYmxpcXVlIC9DaGFyUHJvY3MgMTUgMCBSCi9FbmNvZGluZyA8PCAvRGlmZmVyZW5jZXMgWyAxMTYgL3QgMTIwIC94IF0gL1R5cGUgL0VuY29kaW5nID4+IC9GaXJzdENoYXIgMAovRm9udEJCb3ggWyAtMTAxNiAtMzUxIDE2NjAgMTA2OCBdIC9Gb250RGVzY3JpcHRvciAxMyAwIFIKL0ZvbnRNYXRyaXggWyAwLjAwMSAwIDAgMC4wMDEgMCAwIF0gL0xhc3RDaGFyIDI1NSAvTmFtZSAvRGVqYVZ1U2Fucy1PYmxpcXVlCi9TdWJ0eXBlIC9UeXBlMyAvVHlwZSAvRm9udCAvV2lkdGhzIDEyIDAgUiA+PgplbmRvYmoKMTMgMCBvYmoKPDwgL0FzY2VudCA5MjkgL0NhcEhlaWdodCAwIC9EZXNjZW50IC0yMzYgL0ZsYWdzIDk2Ci9Gb250QkJveCBbIC0xMDE2IC0zNTEgMTY2MCAxMDY4IF0gL0ZvbnROYW1lIC9EZWphVnVTYW5zLU9ibGlxdWUKL0l0YWxpY0FuZ2xlIDAgL01heFdpZHRoIDEzNTAgL1N0ZW1WIDAgL1R5cGUgL0ZvbnREZXNjcmlwdG9yIC9YSGVpZ2h0IDAgPj4KZW5kb2JqCjEyIDAgb2JqClsgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAKNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCAzMTggNDAxIDQ2MCA4MzggNjM2Cjk1MCA3ODAgMjc1IDM5MCAzOTAgNTAwIDgzOCAzMTggMzYxIDMxOCAzMzcgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNgo2MzYgNjM2IDMzNyAzMzcgODM4IDgzOCA4MzggNTMxIDEwMDAgNjg0IDY4NiA2OTggNzcwIDYzMiA1NzUgNzc1IDc1MiAyOTUKMjk1IDY1NiA1NTcgODYzIDc0OCA3ODcgNjAzIDc4NyA2OTUgNjM1IDYxMSA3MzIgNjg0IDk4OSA2ODUgNjExIDY4NSAzOTAgMzM3CjM5MCA4MzggNTAwIDUwMCA2MTMgNjM1IDU1MCA2MzUgNjE1IDM1MiA2MzUgNjM0IDI3OCAyNzggNTc5IDI3OCA5NzQgNjM0IDYxMgo2MzUgNjM1IDQxMSA1MjEgMzkyIDYzNCA1OTIgODE4IDU5MiA1OTIgNTI1IDYzNiAzMzcgNjM2IDg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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x10fdbc0f0>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "%matplotlib inline\n",
    "sym.init_printing()\n",
    "\n",
    "t = sym.symbols('t', real=True)\n",
    "\n",
    "A = .3\n",
    "omA = 3\n",
    "B = .5\n",
    "omB = 5\n",
    "\n",
    "x = A*sym.cos(omA*t) + B*sym.cos(omB*t)\n",
    "\n",
    "sym.plot(x, (t, -5, 5), ylim=[-1.2, 1.2], ylabel=r'$x(t)$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Temporal Shift\n",
    "\n",
    "The temporal shift of a signal $x(t)$ by the time $\\tau$ is a frequently applied operation in signal processing. For instance, to model the propagation of signals from an actuator to a sensor.\n",
    "\n",
    "The temporally shifted signal $x(t)$ is defined as\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = x(t-\\tau)\n",
    "\\end{equation}\n",
    "with $\\tau \\in \\mathbb{R}$. The signal $x(t)$ is\n",
    "\n",
    "* shifted to the right (*delayed*) for $\\tau > 0$\n",
    "* shifted to the left (*leading*) for $\\tau < 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "In order to illustrate the temporal shifting of signals, the construction of a staircase signal by a superposition of shifted [rectangular signals](standard_signals.ipynb#Rectangular-Signal) is considered\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = \\text{rect}\\left(t - \\frac{1}{2} \\right) + \\frac{2}{3} \\cdot \\text{rect}\\left(t-\\frac{3}{2} \\right) + \\frac{1}{3} \\cdot \\text{rect} \\left(t-\\frac{5}{2} \\right)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rect = sym.Heaviside(t + 1/2) - sym.Heaviside(t - 1/2)\n",
    "x = rect.subs(t, t-1/2) + 2/3*rect.subs(t, t-3/2) + 1/3*rect.subs(t, t-5/2)\n",
    "\n",
    "sym.plot(x, (t, -1, 5), ylim=[-0.2, 1.2], ylabel='$x(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Add another step to the beginning of the staircase signal by modifying above example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Temporal Scaling\n",
    "\n",
    "The temporal scaling of a signal $x(t)$ is defined as\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = x(a \\cdot t)\n",
    "\\end{equation}\n",
    "\n",
    "with $a \\in \\mathbb{R}$. The signal $x(t)$ is\n",
    "\n",
    "* stretched for $0 < a < 1$\n",
    "* compressed for $a > 1$\n",
    "* time-reversed and scaled for $a < 0$\n",
    "\n",
    "An application of temporal scaling in signal processing is the adaption of the time scale for standard signals and the modeling of the [Doppler effect](https://en.wikipedia.org/wiki/Doppler_effect)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following example illustrates the temporal scaling of the staircase signal $y(t) = x(a \\cdot t)$ introduced in the previous example. The original $x(t)$ is plotted in gray, the scaled signal $y(t)$ in blue. Here stretching is realized, such that $y(t)$ is twice as long as $x(t)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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   ],
   "source": [
    "a = .5\n",
    "y = x.subs(t, a*t)\n",
    "\n",
    "px = sym.plot(x, (t, -3, 7), ylim=[-0.2, 1.2],\n",
    "              ylabel=r'$y(t)$', show=False, line_color='gray')\n",
    "py = sym.plot(y.subs(a, 1/2), (t, -3, 7),\n",
    "              ylim=[-0.2, 1.2], ylabel=r'$y(t)$', show=False)\n",
    "px.extend(py)\n",
    "px.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Modify above example such that the signal is compressed.\n",
    "* Modify above example such that the signal is scaled and time reversed. What scaling factors `a` lead to  stretching/compression in this context?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Temporal Flipping"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The temporal flipping of a signal $x(t)$ is defined as\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = x(\\tau - t)\n",
    "\\end{equation}\n",
    "\n",
    "for $\\tau \\in \\mathbb{R}$. As $x(\\tau - t) = x(- (t - \\tau))$ the flipping operation can also be represented as a time-reversal of the signal $x(t)$ followed by a shift of $\\tau$ of the reversed signal. For $\\tau = 0$ this results in only a time-reversal of the signal. \n",
    "\n",
    "The temporal flipping operation can be interpreted geometrically as a mirroring of the signal $x(t)$ at the vertical axis $t=\\frac{\\tau}{2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following example illustrates the temporal flipping $y(t) = x(\\tau - t)$ of the staircase signal $x(t)$ introduced before."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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   "source": [
    "tau = -1\n",
    "y = x.subs(t, tau - t)\n",
    "\n",
    "px = sym.plot(x, (t, -5, 5), ylim=[-0.2, 1.2],\n",
    "              ylabel=r'$y(t)$', show=False, line_color='gray')\n",
    "py = sym.plot(y, (t, -5, 5), ylim=[-0.2, 1.2], ylabel=r'$y(t)$', show=False)\n",
    "px.extend(py)\n",
    "px.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Excercise**\n",
    "\n",
    "* For what value $\\tau$ does the signal start at $t=0$?\n",
    "* Realize the temporal flipping by splitting it into two consecutive operations: (i) time-reversal and (ii) temporal shift."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "**Copyright**\n",
    "\n",
    "The notebook is provided as [Open Educational Resource](https://de.wikipedia.org/wiki/Open_Educational_Resources). Feel free to use the notebook for your own educational 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: *Signals and Systems* by Sascha Spors."
   ]
  }
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