21-09-13 L01.ipynb

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    "# 01 - Signals, Systems, and Sequences\n",
    "Lecture 01 - 13 September 2021\n",
    "\n",
    "The first lecture consists mostly of definitions and reminders. These are usually boring and rather cumbersome but are nevertheless needed.\n",
    "\n",
    "Let's start with our first definition: signals.\n",
    "\n",
    "## Signals\n",
    "\n",
    "A signal is a MAP from time to some space:\n",
    "\\begin{align*}\n",
    "    x\\colon\\ &\\ t\\longrightarrow y\\\\\n",
    "    &\\mathcal{R}\\mapsto\\mathcal{R}\n",
    "\\end{align*}\n",
    "\n",
    "It is possible to restrict the domain and the codomain to $\\mathcal{Z} \\subset \\mathcal{R}$ so that we have four types of maps:\n",
    "\n",
    "<!-- I've generated these tables with the help of https://www.tablesgenerator.com/html_tables \n",
    "Trust me, you don't want to deal with tables by hand. -->\n",
    "|   FROM \\ TO   \t|    $\\mathcal{R}$    \t|       $\\mathcal{Z}$      \t|\n",
    "|:-------------:\t|:-------------------:\t|:------------------------:\t|\n",
    "| $\\mathcal{R}$ \t|        ANALOG       \t|         NOT USED         \t|\n",
    "| $\\mathcal{Z}$ \t| DIGITAL<br>(theory) \t| DIGITAL<br>(experiments) \t|\n",
    "\n",
    "\n",
    "## SYSTEM\n",
    "\n",
    "Similarly, systems are MAPs between signals\n",
    "\n",
    "|   FROM \\ TO   \t|             ANALOG                \t|            Digital        \t|\n",
    "|:-------------:\t|:----------------------------:      \t|:----------------------------:\t|\n",
    "|     ANALOG     \t|        Analog System       \t        | Analog to Digital Converter<br>(ADC) |\n",
    "|     DIGITAL   \t| Digital to Analog converter<br>(DAC) \t| Digital system; numeric<br>system; processor |\n",
    "\n",
    "\n",
    "## SEQUENCES\n",
    "\n",
    "Sequences are maps from integer numbers to real or integer numbers:\n",
    "\n",
    "\\begin{align*}\n",
    "    y\\colon\\ n&\\longrightarrow y[n]\\\\\n",
    "    \\mathcal{Z}&\\longmapsto \\mathcal{R} \\textit{ or } \\mathcal{Z}\n",
    "\\end{align*}\n",
    "\n",
    "Two remarkable sequences are the **Delta / Impulse sequence** and the **Step Sequence**.\n",
    "\n",
    "### Delta sequence\n",
    "\n",
    "The delta sequency is defined in analogy to the diract delta and the Kronecker delta:\n",
    "\n",
    "\\begin{align*}\n",
    "    \\delta[n] = \\delta_{n,0} = \\left\\{\\begin{array}{lr}\n",
    "        1 \\text{ if } n=0\\\\\n",
    "        0 \\text{ otherwise}\n",
    "        \\end{array}\\right.\n",
    "\\end{align*}\n",
    "\n",
    "The analogy is straightforward:\n",
    "\n",
    "\\begin{align*}\n",
    "    x(t) &= \\int_{-\\infty}^{+\\infty} x(\\tau) \\delta (t-\\tau) d\\tau\\\\\n",
    "    x[n] &= \\sum_{n_0 = -\\infty}^{+\\infty}x[x_0]\\delta [n-n_0] \n",
    "\\end{align*}\n",
    "\n",
    "### Step Sequence\n",
    "\n",
    "Similarly, the step sequence is defined in analogy to the Heaviside theta:\n",
    "\n",
    "\\begin{align*}\n",
    "    U[n] = \\left\\{\\begin{array}{lr}\n",
    "        1 \\text{ if } n\\geq0\\\\\n",
    "        0 \\text{ otherwise}\n",
    "        \\end{array}\\right.\n",
    "\\end{align*}"
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YPXv2i5bdeeedW8zfYYcd2H333bfqzuU777wzu++++4hk7Y0NoCR12Hve8x5WrlzJTTfdxIMPPsiNN97IUUcdtXl5HYvFQJklSWqKRYsWERG9flC7aNGiXhvDTVasWEFEcO2113Lcccex4447sv/++7Nw4cJ2Rt6CDaAkddjzzz/Pww8/zG233ca6det4xStewbHHHtvvNlUXjKFkliSp3a65u5s5F93Cfud+lzkX3cI1d3e3/ZiLFi1i5syZTJ48eYv5Dz/8ME888US/I3iWLFkCwCWXXMLZZ5/NkiVLOOSQQzjzzDPbmrmVDaCkxqqiaGzYsIHXvva1fOtb3+J1r3sdV1xxBUcffTSrVq3qd7uRKBgf//jH2WOPPZg0aVKfjx/+8IcjllmSpHa65u5uzrt6Kd2r15FA9+p1nHf10rbX877O8t11110A/Z4BXLx4MZMnT+bb3/42xxxzDAcccABveMMbeOKJJ9qWtycbQEmNVFXROPPMM5k6dSqzZ89mxowZfOxjH+Oxxx7jU5/6VL/bjUTBmD9/Pj/60Y9YvHhxn4/eitZQM/fl+uuv58ADD+Swww7jq1/96pD2IUnSxTc+wLr1G7aYt279Bi6+8YG2HrevBnDRokXss88+7Lrrrn1uu2TJEl73utdtcdnG8uXLOeCAA9qStTfbdOxIklQj/RWNUw7vassxly5dymWXXcYnP/nJLeZPmTKFRx99tN9tR6Jg7LLLLkyYMIGddtqpI5l788ILL3DmmWeycOFCxo0bx9y5c3n961/fb7GUJKk3j65et1XzR0J3dzerVq3ikEMOedGym2++mTlz5vS7/eLFi3nf+963xby7776bww47bCRj9sszgJIaqYqicfXVVzNz5ky22267zfPWr1/PsmXLOOigg/rddvHixS+66crWFoyhDAEdbOYTTjiBCy64gDlz5vDSl76UZcuW9Zrhjjvu4OCDD6arq4tJkyZx/PHHc9NNNw36NUiStMmeUyZu1fyR8PzzzwOwbt2W/1644447WLRoEW984xv73PbZZ5/lF7/4BYcffvgW8zvdAHoGUFIj7TllIt29NHvtLBqrVq3i2Wef3WLeFVdcwdq1azn99NP73K6/gnHqqacO+vjz58/nhBNOYNKkSX2u09W15dnPwWZetmwZp556Krfddhuf/exnufbaa3v9dPTRRx/d4hhdXV10d7f/2ktJ0thzzrEHct7VS7cY0TNxwnjOOfbAth1z33335WUvexnnn38+EydOZPr06SxatIgPf/jDnHjiiZx00kl9bvvTn/4UYItmb9WqVaxYscIGUJLarYqiceSRR/IP//APXHzxxZx88snccsstnHXWWXziE5/Y4ktjexqpgjGUIaCDyfzMM88QEbz73e8GijOEU6ZMGfQxJEkaik2XbFx84wM8unode06ZyDnHHti2SzkAIoLvfve7nHPOObz5zW/mueeeY//99+eDH/wg8+fP73fbJUuWMHPmzC0+iL377ruZMGHCgCOBRpINoKRGqqJonH766fzqV7/iS1/6EhdeeCEHH3wwX//61znttNP63a7KgjGYzMuWLeNVr3rV5umlS5fyrne9q9f97bnnnluc8evu7uaII45o3wuQJI1ppxze1dba3Zv99tuPq666aqu3mz9//ouaxGOOOWbzsNJO8RpASY11yuFd3Hbua/jlRa/jtnNf0/YCEhGce+65PPjgg6xZs4bbb799wOYPioJx//33bzFvU8HYdttt2xUXGFzmZcuW8YpXvGLz9NKlSzn00EMBOProo7do+I444giWLVtGd3c3a9eu5Xvf+57fJyhJGrMuvfRSJk2axJ133jnobT7wgQ/0e7nGcNkASlLNDKVYzJ8/v63Foj/Lli3j5S9/OVDc5XP16tXsuuuuZCbLly9nl1122bzuNttsw6c//WnmzZvHnDlzOOuss7wDqCRpTLryyiu59957Wbx48eY6ORjnn38+ixcv5uc//znHH3/8iOdyCKgk1ciVV165+c5iM2bMGPR2F154IWeffTYAe+yxR1uy9eXzn//85ufbbLMNv/zlLwG4//77Oe2005g4ccsb65x00kmcdNJJrFmzZquuR5QkaTTpeWO1wdptt93aWh9tACWpRoZaLKZNm8a0adNGOM3wvOxlL+OSSy6pOoYkSWrhEFBJkiRJaggbQEmSJElqCBtASZIkSR23cePGqiOMWpk55G1tACWNWsP546fRwZ+xJI1NO+64I93d3axfv96/9VspM1m1ahXbb7/9kLb3JjCSRqXx48ezfv36tn8Pnqq1bt06JkyYUHUMSdII22uvvXjyySd56KGHeOyxx6qOA8Dvfve7ITdVnc6x/fbbs9deew1p/zaAkkalKVOm8Nhjj9HV1cW4cQ5mGGsyk3Xr1tHd3c306dOrjiNJGmHjxo1j2rRp3Hvvvbzyla+sOg4At956K4cffnjVMdqewwZQ0qg0depUVqxYwQMPPNDnOqPpk7xOGU1ZJkyYwPTp05k8eXIHU0mSNLbZAEoalcaNG8fee+/d7zpN+SRva5hFkqRmc9yUJEmSJDWEDaAkSZIkNYQNoCRJkiQ1RKUNYERcHhGPR8SyPpbPjYinI2Jx+big0xklSeo066MkqV2qvgnMFcAXgK/3s84PM/PEzsSRJKkWrsD6KElqg0rPAGbmD4CnqswgSVLdWB8lSe0yGq4BPCoilkTE9yLi4KrDSJJUE9ZHSdJWi8ysNkDEvsD1mXlIL8smAxszc21EnAB8LjNn9rLeGcAZANOnT5+1YMGCYedau3YtkyZNGvZ+xkoOMEudc0B9stQlB9QnS11ywNjLMm/evEWZOXuEItXKSNTHct0RrZFj7XdoLOUAs9Q5B9QnS11ygFnalaPf+piZlT6AfYFlg1z3IWBqf+vMmjUrR8LChQtHZD/DVZccmWbpTV1yZNYnS11yZNYnS11yZI69LMBdWXEda9djpOtjjlCNHGu/QyOhLjkyzdKbuuTIrE+WuuTINEtv2l0faz0ENCJ2j4gonx9BMWR1VbWpJEmqlvVRkjRUld4FNCK+BcwFpkbECuAjwASAzPwy8AbgPRHxArAOeFPZ0UqSNGZZHyVJ7VJpA5iZpw+w/AsUt8GWJKkxrI+SpHap9RBQSZIkSdLIsQGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGqLQBjIjLI+LxiFjWx/KIiM9HxPKI+GlEvLLTGSVJ6jTroySpXao+A3gFcFw/y48HZpaPM4AvdSCTNKpcc3c3cy66haXdTzPnolu45u7uqiNJGr4rsD5Kw1Kn+liXLHXJYZZqc1TaAGbmD4Cn+lnlZODrWbgdmBIRe3QmnVR/19zdzXlXL6V79ToAulev47yrl9oESqOc9VEanjrVx7pkqUsOs1Sfo+ozgAPpAh5pmV5RzpMEXHzjA6xbv2GLeevWb+DiGx+oKJGkDrE+Sv2oU32sS5a65DBL9TkiM0d8p1sVIGJf4PrMPKSXZdcDF2Xmj8rpfwc+mJl39VjvDIohMEyfPn3WggULhp1r7dq1TJo0adj7GSs5wCx1zLG0++nNz6dPhMfW/X7ZoV07V5Co+vekVV2y1CUHjL0s8+bNW5SZs0coUq2MRH0sl41ojRxrv0NjKQeYZZM61ce6ZKlLDrN0Jkd/9XGbrY/XUd3AjJbpvcp5W8jMS4FLAWbPnp1z584d9oFvvfVWRmI/YyUHmKWOOc6/6JbNQwXOOvQFPr20+F+6a8pE3vfn1eSq+j1pVZcsdckBZhlDBlUfYeRrZJ1+bnXJUpccYJZN6lQf65KlLjnMUn2Oug8BvQ54W3m3syOBpzNzZdWhpLo459gDmThh/BbzJk4YzznHHlhRIkkdYn2U+lGn+liXLHXJYZbqc1R6BjAivgXMBaZGxArgI8AEgMz8MnADcAKwHPgt8M5qkkr1dMrhxSU/xfjwNXRNmcg5xx64eb6k0cn6KA1PnepjXbLUJYdZqs9RaQOYmacPsDyBv+pQHGlUOuXwLk45vItbb721smGfkkaW9VEavjrVx7pkqUsOs1Sbo+5DQCVJkiRJI8QGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIawgZQkiRJkhrCBlCSJEmSGsIGUJIkSZIaYpu+FkTEqYPY/neZecMI5pEkqfaskZKk0arPBhD4CnAtEP2s8z8Ai5skqWmskZKkUam/BvB7mfmu/jaOiG+OcB5JkkYDa6QkaVTq8xrAzHzLQBsPZh1JksYaa6QkabQa1E1gIuIPI+LNEfG2TY+ROHhEHBcRD0TE8og4t5fl74iIJyJicfl490gcV5KkkWKNlCSNJv0NAQUgIr4B7A8sBjaUsxP4+nAOHBHjgS8CrwVWAHdGxHWZeW+PVb+dme8dzrEkSWoHa6QkabQZsAEEZgMHZWaO8LGPAJZn5oMAEbEAOBnoWdwkSaora6QkaVQZzBDQZcDubTh2F/BIy/SKcl5Pp0XETyPiqoiY0YYckiQNlTVSkjSqxEAfWkbEQuAw4A7guU3zM/OkYR044g3AcZn57nL6rcCrW4eyRMSuwNrMfC4i/hJ4Y2a+ppd9nQGcATB9+vRZCxYsGE40ANauXcukSZOGvZ+xkgPMUuccUJ8sdckB9clSlxww9rLMmzdvUWbOHqFIW62JNXKs/Q6NpRxgljrngPpkqUsOMEu7cvRbHzOz3wfwx709BtpuEPs9CrixZfo84Lx+1h8PPD3QfmfNmpUjYeHChSOyn+GqS45Ms/SmLjky65OlLjky65OlLjkyx14W4K4cZj0azqOJNXKs/Q6NhLrkyDRLb+qSI7M+WeqSI9MsvWl3fRzwGsDM/I+tbDgH605gZkTsB3QDbwLe3LpCROyRmSvLyZOA+9qURZKkrWaNlCSNNn1eAxgR1w+08WDW6UtmvgC8F7iRomj9c2beExEXRsSmoTN/HRH3RMQS4K+Bdwz1eJIkjRRrpCRptOrvDOAfRcR1/SwP4KDhHDwzbwBu6DHvgpbn51EMe5EkqU6skZKkUam/BvDk8r+voLgT2VO9rPP8iCeSJKn+rJGSpFGpzwZw03UNEfFaiqElPwEup7gofaS/70iSpFHDGilJGq0G/B7AzPwwMBO4jOL6gp9HxMcjYv82Z5MkqdaskZKk0WYwXwRP+Wnmr8vHC8BLgKsi4pNtzCZJUu1ZIyVJo8mAXwMREe8H3gY8CXwVOCcz10fEOODnwN+0N6IkSfVkjZQkjTYDNoDALsCpmflw68zM3BgRJ7YnliRJo4I1UpI0qgzmi+A/0s8yv3RWktRY1khJ0mgzqGsAJUmSJEmjnw2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDVEpQ1gRBwXEQ9ExPKIOLeX5dtFxLfL5T+OiH0riClJUsdZIyVJ7VBZAxgR44EvAscDBwGnR8RBPVb7C+A3mXkA8BngE+3Odc3d3cy56BaWdj/NnItu4Zq7u9t9yFrnMEu9c9RJnd6TumSpSw6zjD51rJF1+rnVJUtdcphF0tao8gzgEcDyzHwwM58HFgAn91jnZOBr5fOrgKMjItoV6Jq7uznv6qV0r14HQPfqdZx39dKO/+GqSw6z1DtHndTpPalLlrrkMMuoVasaWaefW12y1CWHWSRtrSobwC7gkZbpFeW8XtfJzBeAp4Fd2xXo4hsfYN36DVvMW7d+Axff+EC7DlnrHGapd446qdN7UpcsdclhllGrVjWyTj+3umSpSw6zSNpakZnVHDjiDcBxmfnucvqtwKsz870t6ywr11lRTv+iXOfJHvs6AzgDYPr06bMWLFgwpExLu5/e/Hz6RHhs3e+XHdq185D2OZpzmKXeOXpau3YtkyZNquTYdXpP6pKlLjnGepZ58+YtyszZI5GtTupWI8fy79Boz2GWgVVZH3uqS5a65ACztCtHf/WxygbwKOCjmXlsOX0eQGb+r5Z1bizX+a+I2Ab4NbBb9hN69uzZeddddw0p05yLbtk8ZOGsQ1/g00u3AaBrykRuO/c1Q9rnaM5hlnrn6OnWW29l7ty5lRy7Tu9JXbLUJcdYzxIRY7UBrFWNHMu/Q6M9h1kGVmV97KkuWeqSA8zSrhz91ccqh4DeCcyMiP0iYlvgTcB1Pda5Dnh7+fwNwC39FbbhOufYA5k4YfwW8yZOGM85xx7YrkPWOodZ6p2jTur0ntQlS11ymGXUqlWNrNPPrS5Z6pLDLJK21jZVHTgzX4iI9wI3AuOByzPznoi4ELgrM68DLgO+ERHLgacoCmDbnHJ4cXlFMU59DV1TJnLOsQdunt8pdclhlnrnqJM6vSd1yVKXHGYZnepWI+v0c6tLlrrkMIukrZaZY+oxa9asHAkLFy4ckf0MV11yZJqlN3XJkVmfLHXJkVmfLHXJkTn2slA0Q5XXntHyGIkaOdZ+h0ZCXXJkmqU3dcmRWZ8sdcmRaZbetLs+VvpF8JIkSZKkzrEBlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhrABlCRJkqSGsAGUJEmSpIawAZQkSZKkhqikAYyIXSLi5oj4efnfl/Sx3oaIWFw+rut0TkmSOs0aKUlqp6rOAJ4L/HtmzgT+vZzuzbrMPKx8nNS5eJIkVcYaKUlqm6oawJOBr5XPvwacUlEOSZLqxhopSWqbqhrA6Zm5snz+a2B6H+ttHxF3RcTtEXFKZ6JJklQpa6QkqW0iM9uz44jvA7v3suh84GuZOaVl3d9k5ouucYiIrszsjoiXArcAR2fmL3pZ7wzgDIDp06fPWrBgwbDzr127lkmTJg17P2MlB5ilzjmgPlnqkgPqk6UuOWDsZZk3b96izJw9QpE6ZjTXyLH2OzSWcoBZ6pwD6pOlLjnALO3K0W99zMyOP4AHgD3K53sADwximyuANwy03qxZs3IkLFy4cET2M1x1yZFplt7UJUdmfbLUJUdmfbLUJUfm2MsC3JUV1LF2PupeI8fa79BIqEuOTLP0pi45MuuTpS45Ms3Sm3bXx6qGgF4HvL18/nbg2p4rRMRLImK78vlUYA5wb8cSSpJUDWukJKltqmoALwJeGxE/B44pp4mI2RHx1XKdlwF3RcQSYCFwUWZa3CRJY501UpLUNttUcdDMXAUc3cv8u4B3l8//Ezi0w9EkSaqUNVKS1E5VnQGUJEmSJHWYDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1hA2gJEmSJDWEDaAkSZIkNYQNoCRJkiQ1RCUNYET8aUTcExEbI2J2P+sdFxEPRMTyiDi3kxklSaqCNVKS1E5VnQFcBpwK/KCvFSJiPPBF4HjgIOD0iDioM/EkSaqMNVKS1DbbVHHQzLwPICL6W+0IYHlmPliuuwA4Gbi37QElSaqINVKS1E51vgawC3ikZXpFOU+SpKazRkqShqRtZwAj4vvA7r0sOj8zrx3hY50BnFFOro2IB0Zgt1OBJ0dgP8NVlxxglt7UJQfUJ0tdckB9stQlB4y9LPuMRJBOG+U1cqz9Do2EuuQAs/SmLjmgPlnqkgPM0pu21se2NYCZecwwd9ENzGiZ3quc19uxLgUuHebxthARd2Vmnxffd0pdcoBZ6pwD6pOlLjmgPlnqkgPMUhejuUbW6edWlyx1yQFmqXMOqE+WuuQAs1SRo85DQO8EZkbEfhGxLfAm4LqKM0mSVAfWSEnSkFT1NRCvj4gVwFHAdyPixnL+nhFxA0BmvgC8F7gRuA/458y8p4q8kiR1ijVSktROVd0F9DvAd3qZ/yhwQsv0DcANHYzWakSHlA5DXXKAWXpTlxxQnyx1yQH1yVKXHGCW2hsFNbJOP7e6ZKlLDjBLb+qSA+qTpS45wCy9aWuOyMx27l+SJEmSVBN1vgZQkiRJkjSCbAD7EBEfjYjuiFhcPk4YeKu2ZzorIjIiplaY4e8i4qfle3JTROxZUY6LI+L+Mst3ImJKFTnKLH8aEfdExMaI6PidoyLiuIh4ICKWR8S5nT5+S47LI+LxiFhWVYYyx4yIWBgR95Y/l/dXmGX7iLgjIpaUWf62qixlnvERcXdEXF9xjociYmn5d+SuKrNoaKyRvR6/FvWxzFKLGll1fSwzWCO3zGGN7DtPY2qkDWD/PpOZh5WPqq5FBIr/YYH/CfyqyhzAxZn58sw8DLgeuKCiHDcDh2Tmy4GfAedVlANgGXAq8INOHzgixgNfBI4HDgJOj4iDOp2jdAVwXEXHbvUCcFZmHgQcCfxVhe/Jc8BrMvMVwGHAcRFxZEVZAN5PccOQOphX/m2t/HbbGjJr5JbqUh+hPjWysvoI1sg+WCP71pgaaQM4enwG+Bug0os2M/OZlskdqShPZt5U3gUP4HaK78CqRGbel5nD/WLloToCWJ6ZD2bm88AC4OQqgmTmD4Cnqjh2jxwrM/Mn5fM1FH/MuyrKkpm5tpycUD4q+X8mIvYCXgd8tYrjS21WeY2sS30ss9SiRlZcH8Ea2VsOa2QvmlYjbQD7995y+MTlEfGSqkJExMlAd2YuqSpDq4j4+4h4BPhzqv2Ec5N3Ad+rOkRFuoBHWqZXUNEf8jqKiH2Bw4EfV5hhfEQsBh4Hbs7MqrJ8luIfyBsrOn6rBG6KiEURcUbVYTRk1sgXZ6lbfQRrpDWyD9bILXyWBtXISr4Goi4i4vvA7r0sOh/4EvB3FD+EvwM+TfFHtIosH6IY2tIR/WXJzGsz83zg/Ig4j+J7qD5SRY5ynfMphjNc2Y4MW5NF9RIRk4B/BT7Q45P5jsrMDcBh5TU434mIQzKzo9eARMSJwOOZuSgi5nby2H34o8zsjohpwM0RcX/56bhqxBq5dTk6WR8Hk6Vcp+010vo4Olkjf6+JNbLRDWBmHjOY9SLiKxTj+TueJSIOBfYDlkQEFMM4fhIRR2TmrzuZpRdXUnwHVVsK3EA5IuIdwInA0dnm7zPZivek07qBGS3Te5XzGi0iJlAUtisz8+qq8wBk5uqIWEhxDUinbwIwBzipvFHH9sDkiPhmZr6lwzkAyMzu8r+PR8R3KIZp2QDWjDVy8Dl60db6OJgsnaqRNa6PYI3slTXyRRpXIx0C2oeI2KNl8vV0/pcRgMxcmpnTMnPfzNyXYvjCK9vV/A0kIma2TJ4M3F9RjuMoTtWflJm/rSJDTdwJzIyI/SJiW+BNwHUVZ6pUFP8KvAy4LzMvqTjLbpvuvhcRE4HXUsH/M5l5XmbuVf4NeRNwS1WFLSJ2jIidNj2nOHNT6V3xtPWskS9Wl/pYZrFGFqyRPVgjX6yJNbLRZwAH8MmIOIxieMtDwF9WmqY+LoqIAynGSD8MzK8oxxeA7ShOjQPcnpmVZImI1wP/AOwGfDciFmfmsZ04dma+EBHvBW4ExgOXZ+Y9nTh2TxHxLWAuMDUiVgAfyczLKogyB3grsLS8rgDgQxXdpXAP4GvlnejGAf+cmZXeXroGplMM84GiBv1TZv7/1UbSEFgjX6wu9RFqUiOrrI9gjeyDNbLeOlIjo80j5yRJkiRJNeEQUEmSJElqCBtASZIkSWoIG0BJkiRJaggbQEmSJElqCBtASZIkSWoIG0BJkiRJaggbQEmSJElqCBtAaRSLiH0j4r6I+EpE3BMRN0XExKpzSZJUNWuk1DsbQGn0mwl8MTMPBlYDp1UbR5Kk2rBGSj3YAEqj3y8zc3H5fBGwb3VRJEmqFWuk1IMNoDT6PdfyfAOwTVVBJEmqGWuk1IMNoCRJkiQ1hA2gJEmSJDVEZGbVGSRJkiRJHeAZQEmSJElqCBtASZIkSWoIG0BJkiRJaggbQEmSJElqCBtASZIkSWoIG0BJkiRJaggbQEmSJElqCBtASZIkSWqI/wMpAEXC+LEPEAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 1080x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
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    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Left: Plot of the delta;\n",
      "Right: Plot of the step sequence.\n"
     ]
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "t = np.arange(-5, 6, 1)\n",
    "delta = np.zeros(t.shape); delta[ t.shape[0]//2] = 1\n",
    "heavi = np.heaviside(t, 1)\n",
    "\n",
    "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 5))\n",
    "\n",
    "ax1.title.set_text('Delta / Impulse Sequence')\n",
    "ax1.scatter(t, delta, marker='o', label=r\"$\\delta[n]=\\delta_{n,0}$\")\n",
    "ax1.set_xlabel('n'); ax1.set_ylabel('y[n]')\n",
    "ax1.grid(True); ax1.legend(prop={'size': 14})\n",
    "ax1.set_xticks(t)\n",
    "ax1.set_ylim((-1, 2))\n",
    "\n",
    "ax2.title.set_text('Step Sequence')\n",
    "ax2.scatter(t, heavi, marker='o', label=r\"$U[n]$\")\n",
    "ax2.set_xlabel('n'); ax2.set_ylabel('y[n]')\n",
    "ax2.grid(True); ax2.legend(prop={'size': 14})\n",
    "ax2.set_xticks(t)\n",
    "ax2.set_ylim((-1, 2))\n",
    "\n",
    "plt.show()\n",
    "print(\"Left: Plot of the delta;\\nRight: Plot of the step sequence.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A sequence can be delayed or anticipated:\n",
    "- Delayed sequence: $y[n] = x[n-n_0],\\ n_0>0 \\in \\mathcal{Z}$;\n",
    "- Anticipated sequence: $y[n] = x[n+n_0],\\ n_0>0 \\in \\mathcal{Z}$.\n",
    "\n",
    "These two operations are essential as they allow us to write all series as a sum of deltas with proper coefficients:\n",
    "\\begin{equation*}\n",
    "x[n] = \\sum_{n_0 = -\\infty}^{+\\infty}x[n_0]\\ \\delta[n-n_0] = (x \\ast \\delta)[n]\n",
    "\\end{equation*}\n",
    "that is a discrete convolution between a sequence and a delta.\n",
    "\n",
    "### Periodic Sequences\n",
    "A sequence $x[n]$ is said to be periodic of period $T$ if and only if (iff) $x[n-T] = x[n]\\ \\forall n \\in \\mathcal{Z}$.\n",
    "\n",
    "### Energy\n",
    "Finally, the energy of a system is defined as\n",
    "\\begin{equation*}\n",
    "\\mathcal{E}(x[n]) = \\sum_{n=-\\infty}^{+\\infty}\\left| x[n] \\right|^2\n",
    "\\end{equation*}\n",
    "where the modulus was introduced to take care of signals $\\mathcal{Z}\\rightarrow \\mathcal{C}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear and Time-Invariant (LTI) Systems\n",
    "\n",
    "As one can immagine, LTI systems are systems that are both linear and time invariant. Let's consider a system:\n",
    "\\begin{align*}\n",
    "    \\mathcal{S}\\colon\\ \\{x[n]\\}&\\longrightarrow \\{y[n]\\}\\\\\n",
    "    x[n]&\\longmapsto y[n]=\\mathcal{S}(x[n])\n",
    "\\end{align*}\n",
    "\n",
    "### Linear Systems\n",
    "\n",
    "A system is linear iff given $y_1[n]=\\mathcal{S}(x_1[n])$ and $y_2[n]=\\mathcal{S}(x_2[n])$ the following relation holds:\n",
    "\n",
    "\n",
    "\\begin{equation*}\n",
    "\\mathcal{S}(\\alpha x_1[n] + \\beta x_2[n]) = \\alpha \\mathcal{S}(x_1[n]) + \\beta \\mathcal{S}(x_2[n]) = \\alpha y_1[n]+\n",
    "\\beta y_2[n]\n",
    "\\end{equation*}\n",
    "\n",
    "for every $\\alpha, \\beta \\in \\mathcal{R}$.\n",
    "\n",
    "```{warning}\n",
    "Saying \"a linear system is one for which the _superposition principle_ holds\" is a tautology because the superposition itself is defined via linearity.\n",
    "```\n",
    "\n",
    "One remarkable consequence of linearity is:\n",
    "\\begin{equation*}\n",
    "\\mathcal{S}(x[n]) = \\mathcal{S}\\left(\\sum_{k=-\\infty}^{+\\infty} x[k] \\delta[n-k]\\right) = \\sum_{k=-\\infty}^{+\\infty} x[k] \\mathcal{S}(\\delta[n-k]) \n",
    "\\end{equation*}\n",
    "\n",
    "that is: **a linear system is fully characterized by its impulse response** $h[n]=\\mathcal{S}(\\delta [n])$.\n",
    "\n",
    "<!-- Linear systems are one of the most important classes of systems because even nonlinear systems can be linearized with apposite techniques. -->\n",
    "\n",
    "### Time-Invariant Systems\n",
    "\n",
    "A system is said to be time invariant iff\n",
    "\\begin{equation*}\n",
    "    y[n] = \\mathcal{S}(x[n]) \\implies y_1[n] = \\mathcal{x[n-n_0]} = y[n-n_0]\n",
    "\\end{equation*}\n",
    "\n",
    "that is: \"to the same inputs correspond the same outputs without regard on when the input was sent\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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